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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
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Outline
• Motivation for 3 dimensional gravity
• Building the theory of Topologically Massive Gravity
• Finding solutions to TMG: Topologically Massive Mechanics
• Outlook
Sabine Ertl (VUT) Topologically Massive Gravity – A theory in 3 Dimensions 16.11.2010 2 / 15
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Why three dimensions?
Three-dimensional gravity is ...
... classically simpler
... physically interesting
... quantum mechanically solvable
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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
Sabine Ertl (VUT) Topologically Massive Gravity – A theory in 3 Dimensions 16.11.2010 3 / 15
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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
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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
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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!
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Building the theory: TMG
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!
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Building the theory: TMG
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!
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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
)
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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’
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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]]
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
)
Sabine Ertl (VUT) Topologically Massive Gravity – A theory in 3 Dimensions 16.11.2010 6 / 15
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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 )
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
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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
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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
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Classification of all Solutions
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)
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
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Classification of all Solutions
Einstein X = 0
BTZ solution
ds2 = a (dx+)2 +1
a(dx−)2 ±
(e2r 1
α+ e−2rα
)dx+ dx− − `2 dr2
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
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Classification of all Solutions
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
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
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Classification of all Solutions
Einstein X = 0
Schrodinger X 6= 0, linear dependence of X , X , X
• 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
Warped X 2 =...X = 0, linear independence of X , X , X
Generic X 2 6= 0 and/or X X 6= 0
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Classification of all Solutions
Einstein X = 0
Schrodinger X 6= 0, linear dependence of X , X , X
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]]
Warped X 2 =...X = 0, linear independence of X , X , X
Generic X 2 6= 0 and/or X X 6= 0
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Classification of all Solutions
Einstein X = 0
Schrodinger X 6= 0, linear dependence of X , X , X
Warped X 2 =...X = 0, linear independence of X , 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 X 2 6= 0 and/or X X 6= 0
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Classification of all Solutions
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
These solutions are neither Einstein, Schrodinger nor warped AdS
In fact: Any generic solution must be non-polynomial in ρ
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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
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Example of the Generic Center I
Naked Singularity – non-anaylitc center
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Example of the Generic Center II
Soliton - no center
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Zooming out ...
... evidence for asymptotic warped AdS behaviour
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... damped oscillations around warped AdS
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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?
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Thank you for your attention
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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.
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