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Noncommutative BTZ Black Holes

KITPC/ITP-CAS Nov. 9. Noncommutative BTZ Black Holes. Collaborartion with M. I. Park, C. Rim, J. H. Yee [arXiv:0710.1362 (hep-th)]. Hyeong-Chan Kim (Yonsei Univ.). Plan. 1.Motivations 2.Three-dimensional noncommutative gravity 3. BTZ black hole with U(1) fluxes

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Noncommutative BTZ Black Holes

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  1. KITPC/ITP-CAS Nov. 9. Noncommutative BTZ Black Holes Collaborartion with M. I. Park, C. Rim, J. H. Yee [arXiv:0710.1362 (hep-th)] Hyeong-Chan Kim (Yonsei Univ.)

  2. Plan 1.Motivations 2.Three-dimensional noncommutative gravity 3. BTZ black hole with U(1) fluxes 4. Properties of the black hole 5. Future directions

  3. Weyl-Moyal Correspondence: 1.Motivations • String motivated space-time uncertainty: • Gravity motivated space-time uncertainty: • Localize extreme precision: gravitational collapse • Spacetime below the Planck scale has no operational meaning • Derive spacetime noncommutativity (Doplicher)

  4. Space-time Non-commutativity Path 2 Path 1

  5. Noncommutativies would modify the short distance behaviors of conventional theories on commutative spaces. • For noncommutative gravities, the smearing(obscuring of the boundary) of the horizons are expected:

  6. ; The precise location of horizon is limited by . For a spherically symmetric case, there is an absolute minimum of the uncertainty in the radial coordinate: We want to get some explicit realizations of smeared horizons in (perturbative) noncommutative black hole solutions; Here, we consider 3-dimensional case. due to Note: The modification is the first order in

  7. Black hole Quantum mechanics Noncommutativity? ?

  8. 2.Three-dimensional noncommutative gravity • 3D gravity in commutative AdS = SL(2,R)xSL(2,R) Chern-Simons gravity. • Noncommutative AdS= GL(2,R)xGL(2,R) Connections: [Achucarro, Townsend; Witten] [Banados, et al]

  9. Action: 2+1 Gravity = Chern-Simons theory Negative cosmological constant: *: Moyal product Equations of motion: Noncommutative Einstein equation: cumbersome to solve for black holes There is an easier way to get the solution !: Seiberg-Witten map

  10. Under the SW map, Chern-Simons action is invariant: Action ( )=Action ( ) + Grandi, Silva (Known solutions) SW map + Remarks: There seems to exist a topological reason for this invariance.

  11. 3. BTZ black hole with U(1) fluxes • In the commutative limit, the equations of motion are decoupled equations: With U(1) fluxes , the action is modified by ( Commutative case ) ( Noncommutative case )

  12. This term is invariant under the Seiberg-Witten Map: = + With appropriate fluxes which decreases rapidly for large . SW map + U(1) fields c are not decoupled anymore; have some nontrivial effect on gravity solutions.

  13. To proceed, we consider Aharonov-Bohm type U(1) potentials: For, (commutative) gravity solution, we consider the BTZ black hole solution: Localized fluxes inside the horizons

  14. SL(2,R) gauge potentials Carlip, et al From the SW map, one obtains solution in the noncommutative gravity are obtained.

  15. Metric of noncommutative gravity is defined by : ,

  16. Splitting of the Killing and apparent horizons: * Apparent horizon: Null hypersurface (r=constant) with has the outer/inner horizons at : equally shifted by 4. Properties of the black hole

  17. * Killing horizon: zero-norm of the Killing vetcor, has the outer/inner horizons at ; not equally shifted Killing horizons do not coincides with apparent horizons in general, except in the non-rotating case.

  18. c

  19. Smearing of the event horizons: Consider a co-rotating frame with the metric * Radial null geodesic: * Near the Killing horizon for or

  20. For non-negative , the outgoing (!) geodesics (real velocity) from the horizon are allowed for . t r

  21. The null signals can escape from (or reach) the horizon in a finite time • Comparison with commutative case: *Infinite time to escape from *Finiteproper time to reach .

  22. The singular behavior of time tnear the horizon is moderated by the noncommutativity effect. The smeared horizon is not so dark, even classically.

  23. Smeared horizon region behaving as a barrier for : Signature change to Euclidean (+++) Imaginary velocity No classical trajectory is allowed Outer (horizon) boundary is hard to penetrate for particles. Cf. (Light or matter) Waves may tunnel when its wavelength is greater than the thickness of the smeared region.

  24. The inner boundary of the smeared region is the trapped surface:Particlescan not escape from this surface. Hawking radiation From Tunneling e+ e- No Hawking radiation From

  25. Inapplicable Hawking temperature by the regular Euclidean geometry: Near the Killing horizon , and : * Periodicity in Ex. Non-rotating case Here, there is no smeared region. Conventional Hawking Temperature:

  26. But, the near horizon regularity is spoiled in the rotating case in general: As the metric blowing up. The conventional definition of Hawking temperature is invalid. This might be a general phenomena for the noncommutivity geometry with the smeared horizons.

  27. 5. Future directions • Higher (or all) orders in ? • Higher dimensional extensions? • Modification of Hawking radiation? We expect some non-thermal spectrum. Thank you very much.

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