topologically massive gravity -- a theory in 3...

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Topologically Massive Gravity –A theory in 3 Dimensions

Sabine Ertl

Institute of Theoretical PhysicsVienna University of Technology

Lunch-Club

16.11.2010

Sabine Ertl (VUT) Topologically Massive Gravity – A theory in 3 Dimensions 16.11.2010 1 / 15

Outline

• Motivation for 3 dimensional gravity

• Building the theory of Topologically Massive Gravity

• Finding solutions to TMG: Topologically Massive Mechanics

• Outlook


Why three dimensions?

Three-dimensional gravity is ...

... classically simpler

... physically interesting

... quantum mechanically solvable





• Einsteins theory in 2+1 dimension probably the simplest model of gravity

• No local degree of freedom

• All perturbative solutions to Einsteins field equations are pure gauge

• Weyl tensor vanishes → Riemann tensor fully determined by Ricci tensor








• Black holes (for Λ < 0): BTZ→ Thermodynamics of BTZ black holes

• For a modified theory: gravitons








• Promising toy model to approach the quantization of GR

• Toy model for quantum gravity

• AdS/CFT correspondence


Building the theory: TMG

Let’s start with Einstein-Gravity in 3 dimensions

IEH =1

16πG

∫d3x√−g R

equations of motion:Rµν = 0

no black holes!



Let’s add a cosmological constant:

IEH+CC =1

16πG

∫d3x√−g (R − 2Λ)

Equations of motion:

Gµν = Rµν −1

2gµνR + Λgµν = 0

• 3 vacuum solutions: Minkowski (Λ = 0), de-Sitter (Λ > 0), anti de-Sitter (Λ < 0)

• For Λ < 0, use parameterization Λ = − 1`2 → R = 6Λ

• BTZ black hole

But still no graviton!



Let’s add a gravitational Chern-Simons term

ITMG =1

16πG

∫d3x√−g[R +

2

`2+

1

2µεαβγ Γρασ

(∂βΓσγρ +

2

3ΓσβτΓτγρ

)]

• Massive propagating degree of freedom and 2 massless boundary gravitons[Li, Song, Strominger [1]]

• CS-Term: maximally chiral

• Equations of motion

Gµν +1

µCµν = 0

Black holes and Gravitons!


Some Aspects of TMG

ITMG =1

16πG

∫d3x√−g[R +

2

`2+

1

2µεαβγ Γρασ

(∂βΓσγρ +

2

3ΓσβτΓτγρ

)]

• Unstable/inconsistent for generic µ → choice of sign of EH-term

• Exception: chiral/logarithmic point µ` = 1:

2 different theories exist (depending on boundary conditions):

• Brown-Henneaux: chiral CFT (unitary) [Li, Song, Strominger [1]]

• Grumiller-Johansson: LCFT (non-unitary) [Grumiller, Johansson [3]]

• Additional boundary terms [Kraus, Larsen [2]]

IGHY +BCC =1

8πG

∫d2x√−g

(K − 1

`2

)


Topologically Massive Mechanics

Finding solutions to TMG is tricky (trivial solutions, no-go theorems), therefore

Simplification: stationary axi-symmetric TMG[Clement 1994 [4] and in 2010 Ertl, Grumiller, Johansson [5]]

Set up: stationary axi-symmetric 3d lineelement + 2d metric → ITMG

ds2 = gµνdxµdxν +e2

X2dρ2 gµν =

(X + YY X−

)µν

with X = (X +, X−, Y )

and det g = X2 = X iXi = X iX jηij = X +X− − Y 2

X2 = 0 : ‘centre’





ITMM =

∫dρ e

(1

2e−2 X2 − 2

`2− 1

2µe−3 εijkX

i X j X k

)

Hamiltonian constraint: G = 12 X

2+ 2

`2 − 1µ εijk X i X j X k = 0

Equations of motion: Xi = − 12µ εijk

(3X j X k + 2X j ˙X k

)





Conserved angular momentum (first integrals to equations of motion )

Ji = εijk X j X k − 14µ

(5X

2+ 12

`2

)Xi + 1

2µ (XX) Xi − 1µ X2Xi

combination with Hamiltonian constraint

εijk J iX j X k = 12 X2X

2 − (XX)2 − 2`2 X2

using equations of motion

XJ = 12µ

((XX)2 − X2X

2)Sabine Ertl (VUT) Topologically Massive Gravity – A theory in 3 Dimensions 16.11.2010 6 / 15




simple but difficult to find analytic solutions → non-existence results:

• |µ`| = 1: Einstein solutions

• |µ`| = 3: null warped black hole


Classification of all Solutions

In general: 6d phase space containing a 4d subspace (Einstein, Schrodinger,Warped AdS) → classification into 4 sectors:

Einstein X = 0

Schrodinger X 6= 0, linear dependence of X , X , X

Warped X 2 =...X = 0, linear independence of X , X , X

Generic X 2 6= 0 and/or X X 6= 0



Einstein X = 0

• Solution to EOM: X = X(0) ρ+ X(2)

• All stationary, axisymmetric solutions of Einstein gravity

• Solutions are locally and asymptotically adS

X =(a,

1

a, ±2

`ρ+ 1

)X =

(a, 0, ±2

`ρ)






Einstein X = 0

BTZ solution

ds2 = a (dx+)2 +1

a(dx−)2 ±

(e2r 1

α+ e−2rα

)dx+ dx− − `2 dr2






Einstein X = 0

BTZ solution

ds2 = −4G`(L du2 + L dv2

)−(`2e2r + 16G 2LLe−2r

)du dv − `2 dr2

L =(r+ + r−)2

16G`L =

(r+ − r−)2

16G`m = L + L j = L− L






Einstein X = 0


• Solutions are: X =(sρ(1∓µ`)/2 + aρ+ b, 0, ± 2

` ρ)

• spacetimes with asymptotic Schrodinger behaviour:

ds2∣∣r→0∼ `2

(±2 dx+ dx− − dr2

r2+ β

(dx+)2

r2z

)z =

1∓ µ`2

• µ` = ∓3: z=2: null warped AdS





Einstein X = 0


ds2 =(sρ(1∓µ`)/2 + aρ+ b

)(dx+)2 ± 4ρ

`dx+ dx− − `2 dρ2

4ρ2

• z < 1: Asymptotic AdS

• z = 1:µ` = 1: 2 logarithmic solutions• Asymptotically AdS: reminiscent of the log-mode [Grumiller, Johansson [3]]

• Marginally violated asymoptotically AdS condition[Skenderis et. all.[6]] [Grumiler, Sachs [7]]





Einstein X = 0



• General solutions X = X(−2) ρ2 + X(0) ρ+ X(2)

• Locally and asymptotically warped (squashed or stretched) AdS[Nutku [8]] [Anninos, Li, Padi, Song, Strominger [9]]

• Warped AdS candidate for stable TMG backgrounds




Einstein X = 0




These solutions are neither Einstein, Schrodinger nor warped AdS

In fact: Any generic solution must be non-polynomial in ρ


Generic Sector

The generic sector is described by the constraints: X 2 6= 0 and/or X X 6= 0Solving for solutions: numerical analysis

Analytic Center

Completely Generic

Einstein

Warped

Schrödinger


Example of the Generic Center I

Naked Singularity – non-anaylitc center


Example of the Generic Center II

Soliton - no center


Zooming out ...

... evidence for asymptotic warped AdS behaviour


... damped oscillations around warped AdS


Outlook

• a lot of new 3D gravity theories: NMG, GMG, MSG, HOMG, BIG

• apply TMM recipe on those novel theories

• CFT’s

• create new 3D theories

• still open questions in TMM• Topography of landscape of solutions• boundary conditions and corresponding asymptotic symmetry group• stability?• Soliton interpretation as finite energy excitations around WAdS?• Soliton asymptotics to AdS or Schrodinger?• Kink solutions?


Thank you for your attention


References

W. Li, W. Song, and A. Strominger, “Chiral Gravity in Three Dimensions,” JHEP 04 (2008) 082,0801.4566.

P. Kraus and F. Larsen, “Holographic gravitational anomalies,” JHEP 01 (2006) 022, hep-th/0508218.

D. Grumiller and N. Johansson, “Instability in cosmological topologically massive gravity at the chiral

point,” JHEP 07 (2008) 134, 0805.2610.

G. Clement, “Particle - like solutions to topologically massive gravity,” Class. Quant. Grav. 11 (1994)

L115–L120, gr-qc/9404004.

S. Ertl, D. Grumiller, and N. Johansson, “All stationary axi-symmetric local solutions of topologically

massive gravity,” 1006.3309.

K. Skenderis, M. Taylor, and B. C. van Rees, “Topologically Massive Gravity and the AdS/CFT

Correspondence,” JHEP 09 (2009) 045, 0906.4926.

D. Grumiller and I. Sachs, “AdS3/LCFT2 – Correlators in Cosmological Topologically Massive Gravity,”

JHEP 03 (2010) 012, 0910.5241.

Y. Nutku, “Exact solutions of topologically massive gravity with a cosmological constant,” Class. Quant.

Grav. 10 (1993) 2657–2661.

D. Anninos, W. Li, M. Padi, W. Song, and A. Strominger, “Warped AdS3 Black Holes,” JHEP 03

(2009) 130, 0807.3040.


http://www.arXiv.org/abs/0801.4566

http://www.arXiv.org/abs/hep-th/0508218


http://www.arXiv.org/abs/gr-qc/9404004





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