Dirac Observables in the 4-Dimensional Phase Space of Ashtekar’s Variables and Spherically Symmetric Loop Quantum Black Holes
In this paper, we study a proposal put forward recently by Bodendorfer, Mele and Münch and García-Quismondo and Marugán, in which the two polymerization parameters of spherically symmetric black hole spacetimes are the Dirac observables of the four-dimensional Ashtekar’s variables. In this model, bl...
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2022-10-01
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| author | Geeth Ongole Hongchao Zhang Tao Zhu Anzhong Wang Bin Wang |
| author_facet | Geeth Ongole Hongchao Zhang Tao Zhu Anzhong Wang Bin Wang |
| author_sort | Geeth Ongole |
| collection | DOAJ |
| description | In this paper, we study a proposal put forward recently by Bodendorfer, Mele and Münch and García-Quismondo and Marugán, in which the two polymerization parameters of spherically symmetric black hole spacetimes are the Dirac observables of the four-dimensional Ashtekar’s variables. In this model, black and white hole horizons in general exist and naturally divide the spacetime into the external and internal regions. In the external region, the spacetime can be made asymptotically flat by properly choosing the dependence of the two polymerization parameters on the Ashtekar variables. Then, we find that the asymptotical behavior of the spacetime is universal, and, to the leading order, the curvature invariants are independent of the mass parameter <i>m</i>. For example, the Kretschmann scalar approaches zero as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>≃</mo><msub><mi>A</mi><mn>0</mn></msub><msup><mi>r</mi><mrow><mo>−</mo><mn>4</mn></mrow></msup></mrow></semantics></math></inline-formula> asymptotically, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>A</mi><mn>0</mn></msub></semantics></math></inline-formula> is generally a non-zero constant and independent of <i>m</i>, and <i>r</i> the geometric radius of the two-spheres. In the internal region, all the physical quantities are finite, and the Schwarzschild black hole singularity is replaced by a transition surface whose radius is always finite and non-zero. The quantum gravitational effects are negligible near the black hole horizon for very massive black holes. However, the behavior of the spacetime across the transition surface is significantly different from all loop quantum black holes studied so far. In particular, the location of the maximum amplitude of the curvature scalars is displaced from the transition surface and depends on <i>m</i>; so does the maximum amplitude. In addition, the radius of the white hole is much smaller than that of the black hole, and its exact value sensitively depends on <i>m</i>, too. |
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| spelling | doaj.art-9c12ce64c3f842deaebf7345150ebb142023-11-24T03:02:03ZengMDPI AGUniverse2218-19972022-10-0181054310.3390/universe8100543Dirac Observables in the 4-Dimensional Phase Space of Ashtekar’s Variables and Spherically Symmetric Loop Quantum Black HolesGeeth Ongole0Hongchao Zhang1Tao Zhu2Anzhong Wang3Bin Wang4GCAP-CASPER, Physics Department, Baylor University, Waco, TX 76798-7316, USAInstitute for Theoretical Physics & Cosmology, Zhejiang University of Technology, Hangzhou 310023, ChinaInstitute for Theoretical Physics & Cosmology, Zhejiang University of Technology, Hangzhou 310023, ChinaGCAP-CASPER, Physics Department, Baylor University, Waco, TX 76798-7316, USACenter for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, Yangzhou 225009, ChinaIn this paper, we study a proposal put forward recently by Bodendorfer, Mele and Münch and García-Quismondo and Marugán, in which the two polymerization parameters of spherically symmetric black hole spacetimes are the Dirac observables of the four-dimensional Ashtekar’s variables. In this model, black and white hole horizons in general exist and naturally divide the spacetime into the external and internal regions. In the external region, the spacetime can be made asymptotically flat by properly choosing the dependence of the two polymerization parameters on the Ashtekar variables. Then, we find that the asymptotical behavior of the spacetime is universal, and, to the leading order, the curvature invariants are independent of the mass parameter <i>m</i>. For example, the Kretschmann scalar approaches zero as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>≃</mo><msub><mi>A</mi><mn>0</mn></msub><msup><mi>r</mi><mrow><mo>−</mo><mn>4</mn></mrow></msup></mrow></semantics></math></inline-formula> asymptotically, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>A</mi><mn>0</mn></msub></semantics></math></inline-formula> is generally a non-zero constant and independent of <i>m</i>, and <i>r</i> the geometric radius of the two-spheres. In the internal region, all the physical quantities are finite, and the Schwarzschild black hole singularity is replaced by a transition surface whose radius is always finite and non-zero. The quantum gravitational effects are negligible near the black hole horizon for very massive black holes. However, the behavior of the spacetime across the transition surface is significantly different from all loop quantum black holes studied so far. In particular, the location of the maximum amplitude of the curvature scalars is displaced from the transition surface and depends on <i>m</i>; so does the maximum amplitude. In addition, the radius of the white hole is much smaller than that of the black hole, and its exact value sensitively depends on <i>m</i>, too.https://www.mdpi.com/2218-1997/8/10/543quantum black holessingularity resolutionasymptotical behavior |
| spellingShingle | Geeth Ongole Hongchao Zhang Tao Zhu Anzhong Wang Bin Wang Dirac Observables in the 4-Dimensional Phase Space of Ashtekar’s Variables and Spherically Symmetric Loop Quantum Black Holes Universe quantum black holes singularity resolution asymptotical behavior |
| title | Dirac Observables in the 4-Dimensional Phase Space of Ashtekar’s Variables and Spherically Symmetric Loop Quantum Black Holes |
| title_full | Dirac Observables in the 4-Dimensional Phase Space of Ashtekar’s Variables and Spherically Symmetric Loop Quantum Black Holes |
| title_fullStr | Dirac Observables in the 4-Dimensional Phase Space of Ashtekar’s Variables and Spherically Symmetric Loop Quantum Black Holes |
| title_full_unstemmed | Dirac Observables in the 4-Dimensional Phase Space of Ashtekar’s Variables and Spherically Symmetric Loop Quantum Black Holes |
| title_short | Dirac Observables in the 4-Dimensional Phase Space of Ashtekar’s Variables and Spherically Symmetric Loop Quantum Black Holes |
| title_sort | dirac observables in the 4 dimensional phase space of ashtekar s variables and spherically symmetric loop quantum black holes |
| topic | quantum black holes singularity resolution asymptotical behavior |
| url | https://www.mdpi.com/2218-1997/8/10/543 |
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