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Although the theory of analog gravity suggests that we can simulate the space-time structure of black holes by using laboratory physical systems, it is difficult to find the analogs for rotating black holes in laboratory systems. In this work, we use a new field form for the optical vortex to study the analogous black hole structure close to the Bañados-Teitelboim-Zanelli (BTZ) black hole. We compare the similarities and differences between massless particles and sound waves by calculating their motions in space-time analogous to BTZ black holes and gravitational BTZ black holes. The effective potential energy values of massless particles and sound waves in both kinds of black hole spacetimes give the same forbidden-zone distributions of energy and angular momentum. The difference is that the classical forbidden area of the BTZ black hole will approach fixed energy values along the radial direction, while the classical forbidden area of the analogous BTZ black hole will be closed along the radial direction. Fortunately, near the event horizon and the ergosphere, the behaviors of massless particles and sound waves are almost the same. From this point of view, we can say that the analogous experimental system can simulate the BTZ black hole very well. In particular, the classically forbidden regions of particles with low energy and high angular momentum are wider in both types of black hole space-time.
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Keywords:
- Bañados-Teitelboim-Zanelli black hole /
- analog black hole /
- optical vortex /
- classically forbidden regions
[1] Unruh W G 1981 Phys. Rev. Lett. 46 1351
Google Scholar
[2] Hawking S W 1974 Nature 248 30
Google Scholar
[3] Lahav O, Itah A, Blumkin A, Gordon C, Rinott S, Zayats A, Steinhauer J 2010 Phys. Rev. Lett. 105 240401
Google Scholar
[4] Nguyen H S, Gerace D, Carusotto I, Sanvitto D, Galopin E, Lemaître A, Sagnes I, Bloch J, Amo A 2015 Phys. Rev. Lett. 114 036402
Google Scholar
[5] Euvé L P, Michel F, Parentani R, Philbin T G, Rousseaux G 2016 Phys. Rev. Lett. 117 121301
Google Scholar
[6] Painlevé P 1921 C. R. Acad. Sci. , Paris 173 677
[7] Gullstrand A 1922 Ark. Mat. Astron. Fys. 16 1
[8] Lemaître G 1933 Ann. Soc. Sci. Brux. A 53 51
[9] Robertson S J 2012 J. Phys. B: At. Mol. Opt. Phys. 45 163001
Google Scholar
[10] Garza P, Kabat D, van Gelder A 2018 Class. Quantum Grav. 35 165009
Google Scholar
[11] Zhang B 2016 Adv. High Energy Phys. 2016 5710625
[12] Visser M 1998 Class. Quantum Grav. 15 1767
Google Scholar
[13] Penrose R, Floyd R M 1971 Nat. Phys. Sci. 229 177
Google Scholar
[14] Fagnocchi S, Finazzi S, Liberati S, Kormos M, Trombettoni A 2010 New J. Phys. 12 095012
Google Scholar
[15] Kroon J A V 2004 Phys. Rev. Lett. 92 041101
Google Scholar
[16] Visser M, Weinfurtner S 2005 Class. Quantum Grav. 22 2493
Google Scholar
[17] Berti E, Cardoso V, Lemos J P S 2004 Phys. Rev. D 70 124006
Google Scholar
[18] Carusotto I, Ciuti C 2013 Rev. Mod. Phys. 85 299
Google Scholar
[19] Braidotti M C, Faccio D, Wright E M 2020 Phys. Rev. Lett. 125 193902
Google Scholar
[20] Ornigotti M, Bar-Ad S, Szameit A, Fleurov V 2018 Phys. Rev. A 97 013823
Google Scholar
[21] Prodanov E M 2014 Class. Quantum Grav. 31 105013
Google Scholar
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Google Scholar
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Google Scholar
[24] Banados M, Teitelboim C, Zanelli J 1992 Phys. Rev. Lett. 69 1849
Google Scholar
[25] Giacomelli L, Liberati S 2017 Phys. Rev. D 96 064014
Google Scholar
[26] Townsend P K, Zhang B 2013 Phys. Rev. Lett. 110 241302
Google Scholar
[27] Zhang B 2013 Phys. Rev. D 88 124017
Google Scholar
[28] Liang C, Gong L, Zhang B 2017 Class. Quantum Grav. 34 035017
Google Scholar
[29] Carlip S 1998 Class. Quantum Grav. 15 3609
Google Scholar
[30] Visser M 1998 Phys. Rev. Lett. 80 3436
Google Scholar
[31] Garay L J, Anglin J R, Cirac J I, Zoller P 2000 Phys. Rev. Lett. 85 4643
Google Scholar
[32] Marino F 2008 Phys. Rev. A 78 063804
Google Scholar
[33] Marino F, Ciszak M, Ortolan A 2009 Phys. Rev. A 80 065802
Google Scholar
[34] Prain A, Maitland C, Faccio D, Marino F 2019 Phys. Rev. D 100 024037
Google Scholar
[35] Boyd R W 2020 Nonlinear Optics (4th Ed.) (New York: Academic Press) pp65–69
[36] Braidotti M C, Prizia R, Maitland C, Marino F, Prain A, Starshynov I, Westerberg N, Wright E M, Faccio D 2022 Phys. Rev. Lett. 128 013901
Google Scholar
[37] Vocke D, Roger T, Marino F, Wright E M, Carusotto I, Clerici M, Faccio D 2015 Optica 2 484
Google Scholar
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Google Scholar
[39] Yan J 2021 Phys. Lett. B 818 136359
Google Scholar
[40] Heckenberg N R, McDuff R, Smith C P, White A G 1992 Opt. Lett. 17 221
Google Scholar
[41] McGloin D, Spalding G C, Melville H, Sibbett W, Dholakia K 2003 Opt. Express 11 158
Google Scholar
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Google Scholar
[43] Farina C, Gamboa J, Segui-Santonja A J 1993 Class. Quantum Grav. 10 L193
Google Scholar
[44] Wilkins D C 1972 Phys. Rev. D 5 814
[45] Chandrasekhar S 1983 The Mathematical Theory of Black Holes (New York: Oxford university press) pp342–347
[46] Cebeci H, Özdemir N, Şentorun S 2016 Phys. Rev. D 93 104031
Google Scholar
[47] Banados M, Henneaux M, Teitelboim C, Zanelli J 1993 Phys. Rev. D 48 1506
[48] Cruz N, Martinez C, Pena L 1994 Class. Quantum Grav. 11 2731
Google Scholar
[49] Solnyshkov D D, Leblanc C, Koniakhin S V, Bleu O, Malpuech G 2019 Phys. Rev. B 99 214511
Google Scholar
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图 1 模拟
$M = 3$ 和$J = 1.3 l$ 旋转BTZ黑洞的流体速度Figure 1. Fluid velocity that analogs to a spinning BTZ black hole with
$M = 3$ and$J = 1.3 l$ .
图 2 涡旋光模拟旋转BTZ黑洞的流体速度(
${\rho _0} = 1.4 \times $ $ {10^5}{\text{ W}}/{{\text{m}}^2}, \sigma = 5{\text{ mm}}, {n_0} = 1.5, \left| {{n_2}} \right| = 4.4 \times {10^{ - 11}}{\text{ }}{{\text{m}}^2}/{\text{W}}$ )Figure 2. Fluid velocity that analogs to a spinning BTZ black hole by the optical vortex (
${\rho _0} = 1.4 \times {10^5}{\text{ W/}}{{\text{m}}^2}, \sigma = 5{\text{ mm}}, $ $ {n_0} = 1.5, \left| {{n_2}} \right| = 4.4 \times {10^{ - 11}}{\text{ }}{{\text{m}}^{\text{2}}}{\text{/W}}$ ).
图 3 BTZ黑洞能层外和能层内的经典禁区
$\left( {M = 3, J = 1.3, l = 1} \right)$ (a)能层外的经典禁区图$ \left( {{r^2} = 9} \right) $ ; (b)能层内的经典禁区图$\left( {{r^2} = 2.9} \right)$ Figure 3. Classically forbidden region of BTZ black hole outside and inside the ergosphere
$\left( {M = 3, J = 1.3, l = 1} \right)$ : (a) The classically forbidden region outside the ergosphere$ \left( {{r^2} = 9} \right) $ ; (b) the classically forbidden region inside the ergosphere$\left( {{r^2} = 2.9} \right)$ .
图 4 BTZ黑洞静界和静界外的经典禁区
$ \left(M=3, J=3, l=1\right) $ (a)${E_ + }$ 和${E_ - }$ 之间的黄色区域是静界处的径向运动的经典禁区, 在这些区域辐射将被禁止发生; (b)静界外角动量${L_ + }$ (蓝线)和${L_ - }$ (红线)两根之外的黄色区域是经典禁区, 图中画的是$E = 1$ 时的${L_ + }$ 和${L_ - }$ 两根, 也就是碰撞参数Figure 4. Classically forbidden region of the BTZ black hole at the static limit and outside the ergosphere
$ \left(M=3, J=3, l=1\right) $ : (a) The yellow area between${E_ + }$ and${E_ - }$ is the classically forbidden area of radial motion at the static limit, where the radiation will be forbidden to occur; (b) the yellow area outside the static limit outside the angular momentum${L_ + }$ (blue line) and${L_ - }$ (red line) is a classically forbidden area, as shown in the figure it is the two roots of${L_ + }$ and${L_ - }$ when$E = 1$ , that is, the impact parameter.
图 5 BTZ黑洞径向的经典禁区 (a)经典禁区位于
${E_ + }$ (蓝线)和${E_ - }$ (红线)之间; 无质量粒子的角动量分别取1, 2, 4, 6$ \left(M=3, J=1.3, l=1\right) $ , BTZ黑洞的内视界、外视界和静界分别位于$ {r}_{-}=0.385, {r}_{+}=1.689, {r}_{\text{s}}=1.732 $ ; (b)角动量L = 2的经典禁区的边界将趋近于2和–2Figure 5. . Classically forbidden region of the BTZ black hole at radial. (a) The classically forbidden region is between
${E_ + }$ (blue line) and${E_ - }$ (red line); the angular momentum of massless particles is 1, 2, 4, 6$ \left(M=3, J=1.3, l=1\right) $ , the inner horizon of the BTZ black hole, the outer horizon and the static limit are respectively located at$ {r}_{-}=0.385, {r}_{+}=1.689, {r}_{\text{s}}=1.732 $ ; (b) the boundary of the classically forbidden region with angular momentum L of 2 will approach to 2 and –2.
图 6 类比黑洞能层外和能层内的经典禁区 (a)能层外的经典禁区图
$\left( {r = 1.6{\text{ mm}}} \right)$ ; (b) 能层内的经典禁区图$\left( {r = 1.5{\text{ mm}}} \right)$ Figure 6. Classically forbidden region of analog black hole outside and inside the ergosphere: (a) The classically forbidden region outside the ergosphere
$\left( {r = 1.6{\text{ mm}}} \right)$ ; (b) the classically forbidden region inside the ergosphere$\left( {r = 1.5{\text{ mm}}} \right)$ .
图 7 类比黑洞静界和静界外的经典禁区 (a)
${E_ + }$ 和${E_ - }$ 之间的黄色区域是静界处的径向运动的经典禁区; (b)静界外角动量${L_ + }$ (蓝线)和${L_ - }$ (红线)两根之外的黄色区域是经典禁区Figure 7. Classically forbidden region of the analog black hole at the static limit and outside the ergosphere: (a) The yellow area between
${E_ + }$ and${E_ - }$ is the classically forbidden area of radial motion at the static limit; (b) the yellow area outside the static limit outside the angular momentum${L_ + }$ (blue line) and${L_ - }$ (red line) is a classically forbidden area.
图 8 类比黑洞径向的经典禁区, 其位于
${E_ + }$ (蓝线)和${E_ - }$ (红线)之间; 无质量粒子的角动量分别取$1 \times {10^{ - 10}}$ ,$2 \times $ $ {10^{ - 10}}$ ,$4 \times {10^{ - 10}}$ ,$8 \times {10^{ - 10}}$ ; 类比BTZ黑洞的内视界、外视界和静界分别位于${r}_{-}=0.36\text{ mm}, {r}_{+}=1.40\text{ mm}, {r}_{\text{s}}= $ $ 1.55\text{ mm}$ Figure 8. Classically forbidden region of the analog black hole at radial. The classical forbidden area is between
${E_ + }$ (blue line) and${E_ - }$ (red line); the angular momentum of massless particles is respectively$1 \times {10^{ - 10}}$ ,$2 \times {10^{ - 10}}$ ,$4 \times $ $ {10^{ - 10}}$ ,$8 \times {10^{ - 10}}$ . The inner horizon of the analog black hole, the outer horizon and the static limit are respectively located at$ {r}_{-}=0.36\text{ mm}, {r}_{+}=1.40\text{ mm}, {r}_{\text{s}}=1.55\text{ mm} $ . -
[1] Unruh W G 1981 Phys. Rev. Lett. 46 1351
Google Scholar
[2] Hawking S W 1974 Nature 248 30
Google Scholar
[3] Lahav O, Itah A, Blumkin A, Gordon C, Rinott S, Zayats A, Steinhauer J 2010 Phys. Rev. Lett. 105 240401
Google Scholar
[4] Nguyen H S, Gerace D, Carusotto I, Sanvitto D, Galopin E, Lemaître A, Sagnes I, Bloch J, Amo A 2015 Phys. Rev. Lett. 114 036402
Google Scholar
[5] Euvé L P, Michel F, Parentani R, Philbin T G, Rousseaux G 2016 Phys. Rev. Lett. 117 121301
Google Scholar
[6] Painlevé P 1921 C. R. Acad. Sci. , Paris 173 677
[7] Gullstrand A 1922 Ark. Mat. Astron. Fys. 16 1
[8] Lemaître G 1933 Ann. Soc. Sci. Brux. A 53 51
[9] Robertson S J 2012 J. Phys. B: At. Mol. Opt. Phys. 45 163001
Google Scholar
[10] Garza P, Kabat D, van Gelder A 2018 Class. Quantum Grav. 35 165009
Google Scholar
[11] Zhang B 2016 Adv. High Energy Phys. 2016 5710625
[12] Visser M 1998 Class. Quantum Grav. 15 1767
Google Scholar
[13] Penrose R, Floyd R M 1971 Nat. Phys. Sci. 229 177
Google Scholar
[14] Fagnocchi S, Finazzi S, Liberati S, Kormos M, Trombettoni A 2010 New J. Phys. 12 095012
Google Scholar
[15] Kroon J A V 2004 Phys. Rev. Lett. 92 041101
Google Scholar
[16] Visser M, Weinfurtner S 2005 Class. Quantum Grav. 22 2493
Google Scholar
[17] Berti E, Cardoso V, Lemos J P S 2004 Phys. Rev. D 70 124006
Google Scholar
[18] Carusotto I, Ciuti C 2013 Rev. Mod. Phys. 85 299
Google Scholar
[19] Braidotti M C, Faccio D, Wright E M 2020 Phys. Rev. Lett. 125 193902
Google Scholar
[20] Ornigotti M, Bar-Ad S, Szameit A, Fleurov V 2018 Phys. Rev. A 97 013823
Google Scholar
[21] Prodanov E M 2014 Class. Quantum Grav. 31 105013
Google Scholar
[22] Mc Caughey E 2016 Eur. Phys. J. C 76 179
Google Scholar
[23] Gillani U A, Saifullah K 2021 Astropart. Phys. 125 102496
Google Scholar
[24] Banados M, Teitelboim C, Zanelli J 1992 Phys. Rev. Lett. 69 1849
Google Scholar
[25] Giacomelli L, Liberati S 2017 Phys. Rev. D 96 064014
Google Scholar
[26] Townsend P K, Zhang B 2013 Phys. Rev. Lett. 110 241302
Google Scholar
[27] Zhang B 2013 Phys. Rev. D 88 124017
Google Scholar
[28] Liang C, Gong L, Zhang B 2017 Class. Quantum Grav. 34 035017
Google Scholar
[29] Carlip S 1998 Class. Quantum Grav. 15 3609
Google Scholar
[30] Visser M 1998 Phys. Rev. Lett. 80 3436
Google Scholar
[31] Garay L J, Anglin J R, Cirac J I, Zoller P 2000 Phys. Rev. Lett. 85 4643
Google Scholar
[32] Marino F 2008 Phys. Rev. A 78 063804
Google Scholar
[33] Marino F, Ciszak M, Ortolan A 2009 Phys. Rev. A 80 065802
Google Scholar
[34] Prain A, Maitland C, Faccio D, Marino F 2019 Phys. Rev. D 100 024037
Google Scholar
[35] Boyd R W 2020 Nonlinear Optics (4th Ed.) (New York: Academic Press) pp65–69
[36] Braidotti M C, Prizia R, Maitland C, Marino F, Prain A, Starshynov I, Westerberg N, Wright E M, Faccio D 2022 Phys. Rev. Lett. 128 013901
Google Scholar
[37] Vocke D, Roger T, Marino F, Wright E M, Carusotto I, Clerici M, Faccio D 2015 Optica 2 484
Google Scholar
[38] Vocke D, Maitland C, Prain A, Wilson K E, Biancalana F, Wright E M, Marino F, Faccio D 2018 Optica 5 1099
Google Scholar
[39] Yan J 2021 Phys. Lett. B 818 136359
Google Scholar
[40] Heckenberg N R, McDuff R, Smith C P, White A G 1992 Opt. Lett. 17 221
Google Scholar
[41] McGloin D, Spalding G C, Melville H, Sibbett W, Dholakia K 2003 Opt. Express 11 158
Google Scholar
[42] Ostrovsky A S, Rickenstorff-Parrao C, Arrizón V 2013 Opt. Lett. 38 34
Google Scholar
[43] Farina C, Gamboa J, Segui-Santonja A J 1993 Class. Quantum Grav. 10 L193
Google Scholar
[44] Wilkins D C 1972 Phys. Rev. D 5 814
[45] Chandrasekhar S 1983 The Mathematical Theory of Black Holes (New York: Oxford university press) pp342–347
[46] Cebeci H, Özdemir N, Şentorun S 2016 Phys. Rev. D 93 104031
Google Scholar
[47] Banados M, Henneaux M, Teitelboim C, Zanelli J 1993 Phys. Rev. D 48 1506
[48] Cruz N, Martinez C, Pena L 1994 Class. Quantum Grav. 11 2731
Google Scholar
[49] Solnyshkov D D, Leblanc C, Koniakhin S V, Bleu O, Malpuech G 2019 Phys. Rev. B 99 214511
Google Scholar
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