Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity
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Citations
Boundary stress tensor and counterterms for weakened AdS3 asymptotic in New Massive Gravity
Magnetised Kerr/CFT correspondence
\mathcal{W} symmetry and integrability of higher spin black holes
Conserved Charges in Asymptotically (Locally) AdS Spacetimes
Conformal Current Algebra in Two Dimensions
References
Mathematical Methods of Classical Mechanics
Role of surface integrals in the Hamiltonian formulation of general relativity
Stability of gravity with a cosmological constant
Related Papers (5)
Frequently Asked Questions (12)
Q2. What is the motivation for considering the spacetime just described?
The motivation for considering the spacetime just described is that it is the analogue of the conic geometry for 2 + 1 gravity with Λ = 0 [12], for which the wedge α Φ 1 is related to total energy and the jump A Φ 0 is related to total angular momentum.
Q3. What is the symmetry group associated with the boundary conditions?
Then the asymptotic symmetries coincide with the Killing vector fields d/dt and d/dφ, and the asymptotic symmetry group associated with these boundary conditions is R x SO(2).
Q4. What is the simplest way to determine the surface deformations?
Then the Hamiltonian,HK]=Jd-x^(x)^(x)-hJK], (3.2)will have well defined variational derivatives, and may be used as the generator of the allowed surface deformations.
Q5. What are the nonzero components of the canonical variables needed for computing expression (3.3?
The only nonzero components of the canonical variables needed for computing expression (3.3) arer κ*4 ,2r*(3.5)which gives- δJ[_ζ] = 4π[(3)<f <Sα - (3)ξφδ(aA)
Q6. What is the definition of the asymptotic symmetries?
The asymptotic symmetries are canonically realized by the "factor group" of surface deformation generators, which is defined by identifying two Hamiltonian generators if they describe the same asymptotic (conformal group) deformation and differ only by a pure gauge deformation.
Q7. What is the simplest way to write the Hamiltonian?
for the boundary conditions and asymptotic symmetries of a gravitation theory to be acceptable, it must be possible to write the Hamiltonian as the usual linear combination of constraints [14]fd"x^(x)Jfμ(x) (3.1)plus an appropriate surface term J[_ζ].
Q8. What is the effect of identifying the points with the points?
By identifying the points (Γ= f ', r = r',φ = φ') with the points (Γ= t' — 2πA, f = r', $ — φ' + 2πα) for all f', r' and φ', this will have the effect of removing a "wedge" of coordinate angle 2π(l — α) and introducing a "jump" of 2πA in coordinate time.
Q9. What is the simplest way to evaluate a deformed surface?
In this case, the central charge K[ξ, η] reduces to the value of the charge J\\_ζ] on the surface deformed by η.To evaluate ./[£] on the deformed surface, the expression (4.11) can be greatly simplified by specializing to r, φ coordinates and using the known asymptotic form of the canonical variables.
Q10. What is the logical connection between the Poisson bracket and the canonical variables?
The critical step in this analysis is to recognize that the "volume" term of the Poisson bracket (5.2) may be calculated by assuming that ξ and η are pure gauge, in which case the charges vanish.
Q11. What is the lapse and shift in the canonical formalism?
From (4.3,4.4), the lapse and shift are determined to be9 \\ (4.9)Kso that the asymptotic behavior of the canonical variables is given by Eqs. (4.4) along withπrr = 0(l/r), π^ = 0(l/r2), π++ = 0(l/r5). (4.10)However, in the canonical formalism, the spacelike surfaces are evolved according to Hamiltonian evolution, which generally differs from Lie transport unless the spatial Einstein equations (3)Gij = Λgij hold.
Q12. What is the lapse and shifts for the spacetime coordinate system?
The lapse and shifts and computed straightforwardly from (2.2); in particular,Γ Γ2 + R2 HI/2 Γ A2R2 Π-1/2αA(r2 + R2) ''r2(a2R2-A2)-A2R2'and, since (3)<f = 0, the component ξr = 0 always.