Status of Birkhoff's Theorem in Polymerized Semiclassical Regime of Loop Quantum Gravity

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The study discusses the status of Birkhoff's theorem in the polymerized semiclassical regime of Loop Quantum Gravity. Topics covered include the collapse of a spherically symmetric cloud of dust, modified Einstein's equations, classical theory elements, polymerization concepts, semiclassical theory formulations, and the Polymerized Einstein Field Equations. The interior solutions and proposals regarding the field equations are also explored at the Seventeenth Marcel Grossmann Meeting in Pescara.

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  1. Status of Birkhoff s theorem in polymerized semiclassical regime of Loop Quantum Gravity Luca Cafaro LC, Jerzy Lewandowski (2024) arXiv:2403.01910 Seventeenth Marcel Grossmann Meeting Pescara, 7-12 July 2024

  2. Introduction Collapse of a spherically symmetric cloud of homogeneous pressureless dust Modified Einstein s equations Oppenheimer-Snyder model (by matching) Seventeenth Marcel Grossmann Meeting Pescara, 7-12 July 2024

  3. Einsteins Equations

  4. Classical theory General spherically symmetric line element: ??2= ???2+(??)2 (?? + ????)2+ ??? 2 (PG coordinates) ?? ???1,???2 ???1,???2 = 2??(?1 ?2) ? = ? = 1 = ??(?1 ?2) Dust Gauge (???? ????? = ?) ? = 1 ??= ?2 Areal Gauge ? ??= 1 + ?(?,?) Seventeenth Marcel Grossmann Meeting Pescara, 7-12 July 2024

  5. Polymerization J. G. Kelly, R. Santacruz, E. Wilson-Ewing. (2020) V. Husain, J. G. Kelly, R. Santacruz, E. Wilson-Ewing. (2022) ? ? ? |??+1 |?? 1 |?? 1-dimensional graph: ? ??+1 ?? 1 ?? Discretization: ? ?? ? Operators: ?? ?? ?? ??= ?? ????(??) ??= ?? 2 ? ??=sin( ????(??)) ?? ?? ??= ?? ????(??) Polymerization: ?? Seventeenth Marcel Grossmann Meeting Pescara, 7-12 July 2024

  6. Semiclassical theory J. G. Kelly, R. Santacruz, E. Wilson-Ewing. (2020) V. Husain, J. G. Kelly, R. Santacruz, E. Wilson-Ewing. (2022) ??2= ??2+(??)2 (?? + ????)2+ ?2? 2 ?2 ? ??? ?2 ? ?? ?? 2 ?? 2 ?? ? ? 2? ???3???2? ??= sin?cos? 2? ? } Polymerized Einstein Field Equations (PEFE) ? = {?,?} ??= ?? ?2 ????3???2? + ??+?? ?2 ?? 1 ? ? = 2?? ? = 8???? ? 4???? ??= ?? ? ??= ? sin?cos? ? Seventeenth Marcel Grossmann Meeting Pescara, 7-12 July 2024

  7. Interior ? 0, ??? = 0 M. Bojowald, J. D. Reyes, R. Tibrewala (2009) K. Giesel, H. Liu, E. Rullit, P. Singh, S. A. Weigl (2023) ? ??= 1 + ?(?,?) ? = ? 8? 3??2+ ? 1 ? 8? 3? + 3 ? 3 { ?? 8??2 ?2 ?? 8??? PEFE ? ???2? = ?2 ?2 ? = ? ?,? ? = ? ??= ??? ? = ?(?) ???2 1 + ?(?)??2+ ?2? 2 ??2= ??2+ (LTB coordinates) 2 ??? ? 8? 3? ? 1 ? 3 ? ?2 = + ?2 ?? 8??? Seventeenth Marcel Grossmann Meeting Pescara, 7-12 July 2024

  8. Interior ? 0, ??? = 0 ???2 1+?(?)??2+ ?2? 2 From ??2= ??2+ ? = ?(?)??? ?(?) = ??? 2 1 2? 2 The Friedmann dust ball: ??2= ??2+ ?2??2+ ?2?? with ??? = ?sin ?? 2 ? ? 8? 3? ? 1 ? 3 ? ?2 = + ?2 ?? 8??? ?3 (pressureless dust) 1 ? Classical Friedmann equation for ?? ( 0) Seventeenth Marcel Grossmann Meeting Pescara, 7-12 July 2024

  9. Static Exterior ? = 0 ?2 ? ?? ?? ? ??= sin?cos? = 0 ? } ??= ?? = ??= 0 1 + ? The metric is fully determined by these 2 expressions 2 ?? 2=2? ? ? ?+? + ? ? 4?2 ?2 ? 2 ??2= ??2+ ?2(?? + ????)2+ ?2? 2 ??2 ?(?)+ ?2? 2 In Schwarzschild coordinates: ??2= ?(?)??2+ 2 ? ? = 1 2? +? ? ?+? ?2 ? 2 Seventeenth Marcel Grossmann Meeting Pescara, 7-12 July 2024

  10. Time dependent Exterior ? = 0 ?2 ? ?? ?? ? ? ??= ??= sin?cos? 0 1 + ?(?,?) ? = ? ? +2? ?2+2? ? { 1 ?2 ?3 ? PEFE ?2+2? ? ???2? = ?2 ?3 ? = ????? or ? = ?(?,?) 2 If ? ?(?,?) then the only line element is given by ? ? = 1 2? ? ?2 ? ?+? ?+ 2 Seventeenth Marcel Grossmann Meeting Pescara, 7-12 July 2024

  11. Oppenheimer-Snyder model (by matching)

  12. Oppenheimer-Snyder model H. Ziaie, Y. Tavakoli (2020) A. Parvizi, T. Pawlowski, Y. Tavakoli, J. Lewandowski (2022) J. Lewandowski, Y. Ma, J. Yang, C. Zhang (2023) ??2= ?(?)??2+??2 VACUUM ?,?,?,? ?(?)+ ?2? 2 2 2 ? ? ???,0 ? ? = 1 2? +? DUST ?,?,?,? ?2 ? 2 ? =? 1 2? 2 ??2= ??2+ ?2??2+ ?2?? with ??? = ?sin ?? ?3 2 ? ? 8? 3? ? 1 ? 3 ? ?2 = + ?2 ?? 8??? ? ? = 4? 3? 1 ? 3 ? ?2 8? 3? ? 3 2 ? ?? 3 ? ?2 + + ?2 ?? 8??? 8??? Seventeenth Marcel Grossmann Meeting Pescara, 7-12 July 2024

  13. Critical mass and horizons 2 2 ? ? ???,0 ? ? = 1 2? +? ?2 ? 2 real solutions of ? ? = 0 (no horizons) If ? ? ?+ ? 32 6 2= 2 4 2 4 + 30 ?2??,0 + ?3??,0 + ?2??,0 ? 21664 96 ???,0 16 16 ???,0 2 real solutions to ? ? = 0 If ? ?+ ? ? 2 13 23 2 2 +1 2 ???,0 6? 1 ???,0 24 ? ?? 2 ?? 2 43 ? + ? ? ? = + 2 2 1 ???,0 8 ? 43 ? + ? ? ?+= 2? Seventeenth Marcel Grossmann Meeting Pescara, 7-12 July 2024

  14. k=0 H. Ziaie, Y. Tavakoli (2020) A. Parvizi, T. Pawlowski, Y. Tavakoli, J. Lewandowski (2022) J. Lewandowski, Y. Ma, J. Yang, C. Zhang (2023) ?2 ?4 ? ? = 1 2? Exact solution to the PEFE with ? = 0 ?+ ? 4 ? = 0, ?+= ? 3 3 2 ? ? 8? 3? 1 ? = ?? + 4??2 ? ? = 4? 3? 1 ? ?? ?? Credits: J. Lewandowski, Y. Ma, J. Yang, C. Zhang Seventeenth Marcel Grossmann Meeting Pescara, 7-12 July 2024

  15. k= -1 2 2 ? 1,0 2 ? ? = 1 2? ? ?2 ? ?+ Exact solution to the PEFE with ? = ? 1,0 2 ?+ 2 ? ? 8? 3? +1 1 ? 3 1 ?2 = ?2 ?? 8??? ? ? = 4? 3? 1 ? 3 1 ?2 8? 3? +1 3 2 ? ?? 3 1 ?2 + + Credits: ?2 ?? 8??? 8??? J. Lewandowski, Y. Ma, J. Yang, C. Zhang Seventeenth Marcel Grossmann Meeting Pescara, 7-12 July 2024

  16. k=1 2 2 ?1,0 2 ? ? = 1 2? ? ?2 ? ? Exact solution to the PEFE with ? = ?1,0 2< 0 ?+ 2 ? ? 8? 3? 1 1 ? 3 1 ?2 = + ?2 ?? 8??? ? ? = 4? 3? 1 ? 3 1 ?2 8? 3? 1 3 2 ? ?? 3 1 ?2 + + Credits: ?2 ?? 8??? 8??? J. Lewandowski, Y. Ma, J. Yang, C. Zhang Seventeenth Marcel Grossmann Meeting Pescara, 7-12 July 2024

  17. Conclusion Two independent methods lead to the very same metric. Static exterior solutions to the Einsten s equation are Schwarzschild-like but depend on two parameters (M and B). There may exist other non-static solutions. If this possibility is ruled out Birkhoff s theorem. Seventeenth Marcel Grossmann Meeting Pescara, 7-12 July 2024

  18. Seventeenth Marcel Grossmann Meeting Pescara, 7-12 July 2024

  19. Classical theory Gravity + Dust: ????+2 ???? 2? ?+ ??) + 4? ??? ?( ?+ ?) ??( ? ? = ?? ?? ?? ???? 2+ ?2 ?2 ???? 2 ???? ?2???? ???? ?? 1 ?= 2?22???? ??+ 4 ?? ?? 2?? 2+ 2???2 ?= 4? ?? 1 ?= 2?????? ?????? ? ??= 4?????? 2? Seventeenth Marcel Grossmann Meeting Pescara, 7-12 July 2024

  20. Classical theory ????+2 ???? 2? ?+ ??) + 4? ??? ?( ?+ ?) ??( ? Gravity + Dust: ? = ?? ?? Dust Gauge (? = ?) ? = 1 ??= ?? Areal Gauge (??= ?2) ? ???? ? ???1,???2 = ??(?1 ?2) ? = ?? ?? ?? ??????? 2+??? ?2 ?? = 4???= 1 +?? ?? 2??? 2? ? Seventeenth Marcel Grossmann Meeting Pescara, 7-12 July 2024

  21. Dust density ? From the Dust Gauge: ?= 4??? By solving the Scalar Constraint: ?= 4???= The density ? is defined by ?= ? ?? ?? ???= ? = 4???? Seventeenth Marcel Grossmann Meeting Pescara, 7-12 July 2024

  22. Quantum theory 1 1 Quantization: ?, ??, ?? ??,??, ? ?? ?? ? ? ? |?? 1 |??+1 |?? 1-dimensional graph: ? ??+1 ?? 1 ?? ?|?? ? =?? ?|?? ? ?? Triad operator: 2 ? ??=sin( ????(??)) Polymerization ?? ?? ??= ?? ????(??) Holonomy: ?? ? = |?? ?+ ?? ??|?? ??= ?? ?|?? ? = 0 If ?? 0 1 Inverse triad: ? = ?|?? ?|?? ? =?? ?|?? ? If ?? 1 ? ?|?? ?? ?? Seventeenth Marcel Grossmann Meeting Pescara, 7-12 July 2024

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