Noncommutative BTZ Black Holes

Collaborartion withM. I. Park, C. Rim, J. H. Yee[arXiv:0710.1362 (hep-th)]

KITPC/ITP-CASNov. 9.

Hyeong-Chan Kim (Yonsei Univ.)

Plan

1.Motivations2.Three-dimensional

noncommutative gravity3. BTZ black hole with U(1) fluxes4. Properties of the black hole5. Future directions

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)

Weyl-Moyal Correspondence:

Path 1 Path 2

Space-time Non-commutativity

• 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:

; 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.

Note: The modification is the first order in

due to

Black hole

Quantum mechanics

Noncommutativity?

?

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]

Action:

Equations of motion:

*: Moyal product

Noncommutative Einstein equation: cumbersome to solve for black holes

There is an easier way to get the solution !: Seiberg-Witten map

2+1 Gravity = Chern-Simons theory

Negative cosmological constant:

• Under the SW map,

Chern-Simons action is invariant: Action ( )=Action ( ) +

SW map

(Known solutions)

Grandi, Silva

Remarks: There seems to exist a topological reason for this invariance.

+

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 )

U(1) fields c are not decoupled anymore; have some nontrivial effect on gravity solutions.

= +

This term is invariant under the Seiberg-Witten Map:

With appropriate fluxes which decreases rapidly for large .

SW map

+

For, (commutative) gravity solution, we consider the BTZ black hole solution:

To proceed, we consider Aharonov-Bohm type U(1) potentials:

Localized fluxes inside the horizons

SL(2,R) gauge potentialsCarlip, et

al

From the SW map, one obtains solutionin the noncommutative gravity are obtained.

Metric of noncommutative gravity is defined by :

,

4. Properties of the black hole

Splitting of the Killing and apparent horizons:

* Apparent horizon: Null hypersurface (r=constant) with

has the outer/inner horizons at

: equally shifted by

* 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.

c

Smearing of the event horizons: Consider

a co-rotating frame with the metric

* Radial null geodesic:

* Near the Killing horizon for

or

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

t

r

• The null signals can escape from (or reach) the horizon in a finite time

• Comparison with commutative case: *Infinite time to escape from

*Finite proper time to reach .

The singular behavior of time t near the horizon is moderated by the noncommutativity effect.

The smeared horizon is not so dark, even classically.

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.

• The inner boundary of the smeared region is the trapped surface: Particles can not escape from this surface.

No Hawking radiationFrom

Hawking radiationFrom

e+ e-

Tunneling

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:

• 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.

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.