Topics in asymptotically flat gravity in 3 and 4 dimensions Glenn - - PowerPoint PPT Presentation
Higher Spin Gravity The Erwin Schrdinger International Institute for Mathematical Physics April 13, 2012 Topics in asymptotically flat gravity in 3 and 4 dimensions Glenn Barnich Physique thorique et mathmatique Universit Libre de
Higher Spin Gravity The Erwin Schrödinger International Institute for Mathematical Physics April 13, 2012
Virasoro algebra.” in preparation.
dimensional asymptotically anti-de Sitter spacetimes.” to appear.
“Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited,” Phys. Rev. Lett. 105 (2010) 111103, 0909.2617. “Aspects of the BMS/CFT correspondence,” JHEP 05 (2010) 062, 1001.1541. “BMS charge algebra,” JHEP 1112 (2011) 105, 1106.0213.
Fefferman-Graham gauge
gµν =
r2
gAB ⇥
2d metric
gAB = r2¯ γAB(xC) + O(1)
flat metric on the cylinder
r t, φ Λ = − 1 l2
existence of general solution integration “constants”
gABdxAdxB = −(r2 + l4 r2 Ξ++Ξ−−)dx+dx− + l2Ξ++(dx+)2 + l2Ξ−−(dx−)2, Ξ++ = Ξ++(x+), Ξ−− = Ξ−−(x−)
BTZ black hole
Ξ±± = 2G(M ± J l )
AdS3 space
M = − 1 8G, J = 0
cannot be taken naively in these coordinates
conformal boundary for flat case: null infinity BMS gauge AdS3 same gauge for asymptotically flat and AdS3 spacetimes fall-offs Minkowski
ds2 = −du2 − 2dudr + r2dφ2
asymptotic symmetries
Lξgrr = 0 = Lξgrφ, Lξgφφ = 0, Lξgur = O(r−1), Lξguφ = O(1), Lξguu = O(1)
general (exact) solution modified bracket linear representation of conformal algebra in bulk spacetime
general solution to EOM
conformal transformation properties asymptotic symmetries transform solutions into solutions
surface charge generators Dirac bracket algebra modes
conventional normalization:
mapping to the plane:
action scaling most general solution solution with flat space solution Penrose rescaling of coordinates limit : null orbifold
alternative scaling well defined limit if limiting metric
appropriate combination for the limit Virasoro algebra contracts to
Virasoro factor: centrally non extended superrotations relation to
similar to contraction between
normalized fields Minkowski background
zero mode solutions in both cases
cosmological solutions
angular defects J
black holes
M angular defects J M angular excess angular excess
3d flat gravity, Chern-Simons formulation invariant metric always exists for Virasoro CS co-adjoint vs adjoint extension of 3d flat gravity
cosmological constant same inner product no such tensor, AdS deformation does not survive extension invariance of metric implies completely skew, Jacobi implies invariant under co-adjoint action extended theory “exotic deformation” survives on its own related to to be studied further: asymptotics, boundary theory, solutions, 1-loop effects ...
BMS ansatz
Sachs: unit sphere
gABdxAdxB = r2¯ γABdxAdxB + O(r) ¯ γAB = e2ϕ
0γAB 0γABdxAdxB = dθ2 + sin2 θdφ2
ζ = eiφ cot θ 2, ¯ γABdxAdxB = e2e
ϕdζd¯
ζ
Riemann sphere determinant condition
det gAB = r4 4 e4e
ϕ
fall-off conditions
u r
dθ2 + sin2 θdφ2 = P −2dζd¯ ζ, P(ζ, ¯ ζ) = 1 2(1 + ζ ¯ ζ), ˜ ϕ = ϕ − ln P
β = O(r−2), U A = O(r−2), V/r = −1 2 ¯ R + O(r−1) xA = ⇢ θ, φ ζ, ¯ ζ gµν = −e−2β −e−2β − V
r e−2β
−U Be−2β −U Ae−2β gAB
Minkowski
u = t − r
ηµν = −1 −1 −1 r2 r2 sin2 θ
ζ = cot θ 2eiφ
asymptotic symmetries
general solution conformal Killing vectors of the sphere
Lξgrr = 0, LξgrA = 0, LξgABgAB = 0, Lξgur = O(r−2), LξguA = O(1), LξgAB = O(r), Lξguu = O(r−1) Y A = Y A(xB) T = T(xB)
generators for supertranslations algebra
[(Y1, T1), (Y2, T2)] = ( Y , T)
spacetime vectors with modified bracket form linear representation of
bms4
standard GR choice: restrict to globally well-defined transformations generators of Lorentz algebra
Y A
= Y B
1 ∂BY A 2 − Y B 1 ∂BY A 2 ,
= Y A
1 ∂AT2 − Y A 2 ∂AT1 + 1
2 (T1∂AY A
2 − T2∂AY A 1 )
SL(2, C)/Z2 ' SO(3, 1)
⇤ ⌅ ⇥ ξu = f, ξA = Y A + IA, IA = −f,B ⇧ ⇥
r
dr(e2βgAB), ξr = − 1
2r( ¯
DAξA − f,BU B + 2f∂uϕ), ˙ f = f ˙ ϕ + 1
2 ψ ⇐
⇒ f = eϕ T + 1
2
⇤ u du⇥eϕψ ⇥ ,
ψ = ¯ DAY A
Sachs 1962
CFT choice : allow for meromorphic functions on the Riemann sphere
Y ζ = Y ζ(ζ), Y
¯ ζ = Y ¯ ζ(¯
ζ)
solution to conformal Killing equation generators
ln = −ζn+1 ∂ ∂ζ , ¯ ln = −¯ ζn+1 ∂ ∂¯ ζ , n ∈ Z Tm,n = ζm¯ ζn, m, n ∈ Z
commutation relations
[lm, ln] = (m − n)lm+n, [¯ lm, ¯ ln] = (m − n)¯ lm+n, [lm, ¯ ln] = 0, [ll, Tm,n] = (l + 1 2 − m)Tm+l,n, [¯ ll, Tm,n] = (l + 1 2 − n)Tm,n+l.
Poincaré subalgebra
l−1, l0, l1, ¯ l−1, ¯ l0, ¯ l1, T0,0, T1,0, T0,1, T1,1,
superrotations supertranslations
free data u dependence fixed through evolution equation integration “constants free u dependence news tensor plays no role asymptotically unit sphere
bms4 transformations Interpretation and consequences: work in progress field dependent Schwarzian derivative: Lie algebra Lie algebroid
asymptotic charge : non integrable due to the news
δ /Qξ[δX, X] = δ (Qs[X]) + Θs[δX, X] , {Qs1, Qs2}∗ [X] = (−δs2)Qs1[X] + Θs2[−δs1X, X] .
Proposal : “Dirac” bracket Proposition :
K[s1,s2],s3 − δs3Ks1,s2 + cyclic (1, 2, 3) = 0.
generalized cocycle condition
{Qs1, Qs2}∗ = Q[s1,s2] + Ks1,s2,
if one can integrate by parts
supertranslations :
QTm,n,0[X Kerr] = 2M G Im,n, Im,n = 1 4π Z d2Ω 1 1 + ζ ¯ ζ ζm¯ ζn . Im,n = δm
n I(m)
QT =1,Y =0[X Kerr] = M G , I(m) = 1 4 Z 1
−1
dµ (1 + µ)m (1 − µ)m−1
divergences for proper supertranslations ! superrotations :
Q0,lm[X Kerr] = −δm iaM 2G . ∂φ = −i(l0 − ¯ l0) QT =0,Y φ=1,Y θ=0[X Kerr] = −Ma G
central charges :
K(0,lm),(0,ln)[X Kerr] = 0 = K(0,¯
lm),(0,¯ ln)[X Kerr] = K(0,lm),(0,¯ ln)[X Kerr],
K(0,ll),(Tm,n,0)[X Kerr] = a l(l − 1)(l + 1) 16G Jm+l,n , K(0,¯
ll),(Tm,n,0)[X Kerr] = a l(l − 1)(l + 1)
16G Jm,n+l ,
Jm,n = δm
n J(m)
J(m) = 2 Z 1
−1
dµ (1 + µ)m− 3
2
(1 − µ)m+ 1
2 ,
form in-line with extremal Kerr/CFT correspondence, but divergences ! problem: one cannot integrate by parts if there are poles there are no poles for but then way out (i) define the analog of charge fields to regularize the divergences in the charges (ii) take correctly into account the boundary contributions to correct the central charges Green’s function for
4d gravity is dual to an extended conformal field theory to be done:
Penrose, Les Houches 1963
scattering theory particles as UIRREPS for
between and
angular momentum problem in GR: Lorentz = bms4(old)/supertranslations bms4(new)/supertranslations = Virasoro
Geroch, Asymptotic structure of spacetime, 1977
4 conditions needed to fix rotations Lorentz = Poincaré /translation infinite # conditions needed to fix rotations infinite # conditions needed to fix infinite #
base space
M A
ρA
TM & . M
Lie algebroid vector bundles and
A TM
bundle map, “anchor”
ρA : A → TM [·, ·]A
Lie bracket on Γ[A] Lie algebra homomorphism + Leibniz rule
[α, fβ]A = f[α, β]A + (ρA(α)f)β α, β ∈ Γ[A], f ∈ C∞(M)
local coordinates
A 3 f = f α(φ)eα M : φi ρA(f) = f αRi
α
∂ ∂φi = δf [f1, f2]A =
αβ(φ)(f α 1 , f β 2 ) + δf1f γ 2 − δf2f γ 1 )eγ
GR: fields are Riemannian metrics satisfying Einstein’s equation quantitative control on functional aspects: jet-spaces & variational bicomplex gauge theories
M φi
s(x)
solutions to underdetermined EL
TM
linearized solutions
δφi
s(x)
generating set of irreducible Noether operators
R†i
α
R+i
α [ δL
δφi ] = 0 δL δφi ≈ 0
determines structure functions and algebra A field dependent gauge parameters on
M
gauge parameters are metric dependent vector fields isotropy Lie algebra : dynamical Killing vectors at a particular solution asymptotic context: physically meaningful sub-Lie algebroids of the gauge algebroid for GR reduce to action Lie algebroids involving the Virasoro algebras bracket: derived bracket from antibracket in BV description
Asymptotically flat spacetimes & symmetries
Gravitational AdS3/CFT2 & Kerr/CFT
Holography at null infinity in 3 & 4 dimensions
This work based on
spacetimes at null infinity revisited. Phys. Rev. Lett., 105(11):111103, Sep 2010.
submitted to Proceedings of XXIX Workshop on Geometric Methods in Physics Białowieża - 27.06-03.07.2010
010 (2010) arXiv1102.4632.
algebra.” in preparation
asymptotically anti-de Sitter spacetimes,” 1204.3288.