Topologically Massive Gravity – A theory in 3 Dimensions

Topologically Massive Gravity – A theory in 3 Dimensions

Sabine Ertl

Institute of Theoretical Physics Vienna University of Technology

Lunch-Club

16.11.2010

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

Why three dimensions?

Three-dimensional gravity is ...

... classically simpler

• 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

... physically interesting

... quantum mechanically solvable

Why three dimensions?

Three-dimensional gravity is ...

... classically simpler ... physically interesting

• Black holes (for Λ<0): BTZ

→Thermodynamics of BTZ black holes

• For a modified theory: gravitons

... quantum mechanically solvable

Why three dimensions?

Three-dimensional gravity is ...

... classically simpler ... physically interesting

... quantum mechanically solvable

• 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 Z

d³x√

−g R

equations of motion:

Rµν = 0

no black holes!

Building the theory: TMG

Let’s add a cosmological constant:

IEH+CC= 1

16πG Z

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 =−_`¹2 → R = 6Λ

• BTZ black hole

But still no graviton!

Building the theory: TMG

Let’s add a gravitational Chern-Simons term ITMG= 1

16πG Z

d³x√

−gh R+ 2

`² + 1

2µε^αβγΓ^ρ_ασ ∂βΓ^σ_γρ+2

3Γ^σ_βτΓ^τ_γρi

• Massive propagating degree of freedom and 2 massless boundary gravitons

[Li, Song, Strominger [1]]

• CS-Term: maximally chiral

• Equations of motion

G_µν+ 1

µC_µν= 0

Black holes and Gravitons!

Some Aspects of TMG

ITMG= 1 16πG

Z d³x√

−gh R+ 2

`² + 1

2µε^αβγΓ^ρ_ασ ∂βΓ^σ_γρ+2

3Γ^σ_βτΓ^τ_γρi

• Unstable/inconsistent for genericµ → 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 Z

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=gµνdx^µdx^ν+ e²

X²dρ² gµν =

X⁺ Y Y X⁻

µν

with X= (X⁺,X⁻,Y)

and detg =X²=XⁱXi =XⁱX^jηij =X⁺X⁻−Y² X²= 0 : ‘centre’

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]]

I_TMM= Z

dρe 1

2e⁻²X˙²− 2

`² − 1

2µe⁻³_ijkXⁱX˙^jX¨^k

Hamiltonian constraint: G = ¹₂X˙²+_`²₂ −_µ¹_ijkXⁱX˙^jX¨^k = 0 Equations of motion: X¨i=−_2µ¹ ijk 3 ˙X^jX¨^k+ 2X^jX˙¨^k

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]]

Conserved angular momentum (first integrals to equations of motion ) J_i =_ijkX^jX˙^k −_4µ¹ 5 ˙X²+¹²_`2

X_i+_2µ¹ (XX) ˙˙ X_i−_µ¹X²X¨_i combination with Hamiltonian constraint

_ijkJⁱX^jX˙^k =¹₂X²X˙²−(XX)˙ ²−_`²₂X² using equations of motion

XJ=_2µ¹ (XX)˙ ²−X²X˙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]]

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, Schr¨odinger, Warped AdS)→classification into 4 sectors:

Einstein

Schr¨ odinger

Warped

X = 0, linear independence ofX,X˙,X¨

Generic

Classification of all Solutions

Einstein

• 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

Schr¨ odinger

Warped

X = 0, linear independence ofX,X˙,X¨

Generic

Classification of all Solutions

Einstein

ds²=a(dx⁺)²+1

a(dx⁻)²± e^2r1

α+e^−2rα

dx⁺dx⁻−`²dr²

Schr¨ odinger

Warped

X = 0, linear independence ofX,X˙,X¨

Generic

Classification of all Solutions

Einstein

ds²=−4G` Ldu²+ ¯Ldv²

− `²e^2r+ 16G²L¯Le^−2r

dudv−`²dr² L= (r++r−)²

16G`

¯L= (r+−r−)²

16G` m=L+ ¯L j=L−L¯

Schr¨ odinger

Warped

X = 0, linear independence ofX,X˙,X¨

Generic

Classification of all Solutions

Einstein

Schr¨ odinger

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

• spacetimes with asymptotic Schr¨odinger behaviour:

ds²

r→0∼`²±2 dx⁺dx⁻−dr²

r² +β (dx⁺)² r^2z

z = 1∓µ`

2

• µ`=∓3: z=2: null warped AdS

Warped

X = 0, linear independence ofX,X˙,X¨

Classification of all Solutions

Einstein

Schr¨ odinger

(dx⁺)²±4ρ

` dx<sup>+</sup>dx<sup>−</sup>−`²dρ² 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]]

Warped

X = 0, linear independence ofX,X˙,X¨

Generic

Classification of all Solutions

Einstein

Schr¨ odinger

Warped

X = 0, linear independence ofX,X˙,X¨

• 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

Generic

Classification of all Solutions

Einstein

Schr¨ odinger

Warped

X = 0, linear independence ofX,X˙,X¨

Generic

These solutions are neither Einstein, Schr¨odinger nor warped AdS In fact: Any generic solution must be non-polynomial inρ

Generic Sector

The generic sector is described by the constraints: ¨X²6= 0 and/or ˙XX¨ 6= 0 Solving 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 Schr¨odinger?

• Kink solutions?

Thank you for your attention

References

W. Li, W. Song, and A. Strominger, “Chiral Gravity in Three Dimensions,”JHEP04(2008) 082, 0801.4566.

P. Kraus and F. Larsen, “Holographic gravitational anomalies,”JHEP01(2006) 022,hep-th/0508218.

D. Grumiller and N. Johansson, “Instability in cosmological topologically massive gravity at the chiral point,”JHEP07(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,”JHEP09(2009) 045,0906.4926.

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

JHEP03(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 AdS3Black Holes,”JHEP03 (2009) 130,0807.3040.