Asymptotically flat spacetimes at null infinity revisited

Transcription

1 QG2. Corfu, September 17, 2009 Asymptotically flat spacetimes at null infinity revisited Glenn Barnich Physique théorique et mathématique Université Libre de Bruxelles & International Solvay Institutes So thank you for the occasion to speak at this conference. Today I would like to talk about how 2 dimensional conformal field theory arises directly in the context of classical 4 dimensional GR.

2 Overview Classical gravitational aspects of AdS3/CFT2 correspondence 3d flat case, null infinity: BMS3/CFT1 correspondence 4d flat case, null infinity: BMS4/CFT2 correspondence work done in collaboration with C. Troessaert As has been done previously in this school, in order to motivate the results in the 4 dimensional case, I will start with gravity in 3 dimensions. More precisely, since this might not be well-known in this audience, I will start by reviewing some of the purely classical and gravitational aspects of the ADS3/CFT2 correspondence. In this case, boundary conditions are given at spatial infinity. Then I will move to the flat case and consider null infinity. The asymptotic symmetry algebra in this case is well known and involves as a subalgebra the 1 copy of the Virasoro algebra. Finally, I will consider the 4d case and show that the asymptotic symmetry algebra, the so-called BMS algebra as defined in the early 60 s, can be taken to involve as a subalgebra a left and right copy of the Virasoro algebra in a completely natural way and will derive the first consequences of this observation.

3 AdS3/CFT2 Asymptotic symmetries Fefferman-Graham ansatz g µν = ( l 2 r g AB ) g AB = r 2 γ AB (x C )+O(1) 2d metric γ AB conformal to flat metric on the cylinder γ AB = e 2ϕ η AB η AB dx A dx B = dτ 2 + dφ 2, τ = t l, ϕ = ϕ(xa ) asymptotic symmetries L ξ g rr =0=L ξ g ra, L ξ g AB = O(1), general solution determined by conformal Killing vector { ξ r = 1 2 ψr, ξ A = Y A + I A, I A = l2 2 Bψ r dr r Y A of η AB g AB = l2 4r 2 γ AB B ψ + O(r 4 ). ψ = D A Y A The Fefferman-Graham ansatz consists in defining an asymptotically AdS3 spacetime by a metric of the following form, where the dynamical variables are all encoded in a 2d metric whose leading order as one approaches spatial infinity is conformal to the flat metric on the cylinder. This class of spacetimes includes besides anti-de Sitter space also the BTZ black hole for instance. Asymptotic symmetries are defined as those transformations that leave the FG form invariant. The generators can then be shown to be determined by a conformal Killing vector of the boundary metric and thus of the flat metric on the cylinder. Note that the subleading orders of these vectors also depend explicitly on the metric. In some sense, these transformations constitute the residual global symmetries after the Fefferman- Graham gauge fixation.

4 AdS3/CFT2 Asymptotic symmetries modified bracket [ξ 1, ξ 2 ] µ M =[ξ 1, ξ 2 ] µ δ g ξ 1 ξ µ 2 + δg ξ 2 ξ µ 1 faithful representation of conformal algebra [ξ 1, ξ 2 ] r M = 1 2 ψr, [ξ 1, ξ 2 ] A M = Ŷ A + ÎA, light-cone coordinates x ± = τ ± φ, Ŷ A =[Y 1,Y 2 ] A, 2 ± = τ ± φ, ψ = DA Ŷ A γ ABdx A dx B = e 2ϕ dx + dx Y ± (x ± ) ± = n Z c n ±l ± n, l ± n ± = (x ± ) n+1 ±, [l ± m,l ± n ]=(m n)l ± m, [l ± m,l n ]=0 include Weyl rescalings of boundary metric L ξ g rr =0=L ξ g ra, L ξ g AB =2ωg AB + O(1) direct sum with abelian algebra of Weyl rescalings (Ŷ, ω) = [(Y 1, ω 1 ), (Y 2, ω 2 )] Ŷ A = Y B 1 B Y A 2 Y B 2 B Y A 1, ω =0 If one then introduces the following modified bracket, which I believe is new and takes into account this dependence, one finds that the spacetime vectors solving the asymptotic Killing equations form a faithful representation of the conformal algebra. In this case here, the spacetime vectors form a faithful representation of the conformal algebra in the usual bracket to leading order and the modification is not so essential, but in the cases to be discussed below, this is no longer necessarily the case. For later use and to fix ideas, let me also shortly flash the explicit form of the conformal algebra in light-cone coordinates, where we have two copies of the Witt algebra. One can also generalize the above considerations by including the conformal rescalings of the boundary metric,or in other words, shifts in phi. The result is then the direct sum of the conformal algebra with the abelian algebra of Weyl rescalings. My point here is that both transformations coexist and really play a different role as I will now show.

5 AdS3/CFT2 general solution Solution space g AB = r 2 γ AB + γ AB, γ AB = n r 2i (i)γ AB, i=0 existence of closed form for ϕ =0 g AB dx A dx B = (r 2 + l4 r 2 Ξ ++Ξ )dx + dx + l 2 Ξ ++ (dx + ) 2 + l 2 Ξ (dx ) 2, BTZ black hole Ξ ±± =2G(M ± J l ) coordinates on solution space Ξ ++ = Ξ ++ (x + ), Ξ = Ξ (x ) ϕ(x +,x ) conformal transformations δ Y +,Y,ωΞ ±± = Y ± ± Ξ ±± +2 ± Y ± Ξ ±± ±Y ± δ Y +,Y,ωϕ = ω Using the FG ansatz, one can prove that the general solution to Einstein s equation with negative cosmological constant can be expanded as a power series with even powers of 1/r by starting from the boundary metric at r^2. One can then recursively solve the equations in terms of the boundary metric, up to integrations constants that arise in the process. In fact the recursion stops at order r^[-2}. Let me give it when the boundary metric is flat phi=0. It involves two arbitrary functions, one of x^+ and the other of x^-. In particular, for instance, the BTZ BH corresponds to the case when these functions are constants. The general solution is slightly more complicated, but not very much so and involves in addition the curvature of the boundary metric, in this parameterization in the form of derivatives of phi. Coordinates on solution space are thus provided by two functions of 1 variable and one functions of 2 variables. The extended transformation that we have considered before transform solutions to solutions. They thus act on solution space. Explicitly one finds that the arbitrary functions transform exactly like the components of an energy-momentum tensor of a 2 dimensional conformal field theory, besides the conformal weight which is 2,0 and 0,2 respectively, there is an inhomogeneous term involving the third derivative of the transformation parameter.

6 AdS3/CFT2 Charge algebra Hamiltonian approach ξ surface charge generators, Dirac algebra centrally extended charge representation of algebra of asymptotic Killing vectors covariant version Q ξ [g ḡ, ḡ] = 1 8πG 2π 0 dφ (Y + Ξ ++ + Y Ξ ) Q ξ1 [L ξ2 g, ḡ] Q [ξ1,ξ 2 ] M [g ḡ, ḡ]+k ξ1,ξ 2, K ξ1,ξ 2 = Q ξ1 [L ξ2 ḡ, ḡ] = 1 8πG 2π 0 dφ ( φ Y τ 1 2 φy φ 2 φy τ 2 2 φy φ 1 ) modes in combination with Cardy formula gives a microscopic derivation of the Bekenstein-Hawking entropy of BTZ black hole What has also been done in the Hamiltonian approach to the problem by Brown & Henneaux is the construction of the generators of the conformal transformations in terms of surface charges. Furthermore, they have computed the Dirac brackets of these surface charges and found that they form a centrally extended representation of the conformal algebra. What is extremely interesting about this computation is that in a conformal field theory, the value of the central charge contains information about the number of states in the theory. This has been used by Strominger to give a microscopic derivation of Bekenstein-Hawking entropy of the BTZ black hole.

7 BMS3/CFT1 ansatz for asymptotically flat metrics g µν = Asymptotic symmetries e 2β Vr 1 + r 2 e 2ϕ U 2 e 2β r 2 e 2ϕ U e 2β 0 0 r 2 e 2ϕ U 0 r 2 e 2ϕ Minkowski spacetime ds 2 = du 2 2dudr + r 2 dφ 2 u = t r fall-off conditions β = O(r 1 ), U = O(r 2 ) V = 2r 2 u ϕ + O(r) asymptotic symmetries L ξ g rr =0=L ξ g rφ, L ξ g φφ =0, L ξ g ur = O(r 1 ), L ξ g uφ = O(1), L ξ g uu = O(1) exact solution involves 2 arbitrary functions on the circle Y = Y (φ), T = T (φ) spacetime vector form faithful representation of bms 3 algebra [(Y 1,T 1 ), (Y 2,T 2 )] = (Ŷ, T ) Ŷ = Y 1 φ Y 2 (1 2), T = Y1 φ T 2 + T 1 φ Y 2 (1 2) In the flat 3d case, the analog of the FG ansatz for asymptotically flat metrics at null infinity is the following. It is motivated by the geometrical considerations of the wellstudied four dimensional case. Minkowski space corresponds to phi,beta and U equal to zero with V=-r. r goes to infinity at u fixed, and the angle fixed goes along a light ray. Again, the asymptotic symmetries are generated by the vector fields that leave this form of the metric invariant. These equations can be exactly solved. The precise form is not so important. It contains two arbitrary functions on the circle Y and T with all subleading terms determined in terms of the metric. In the modified bracket, these vectors form a faithful representation of the so-called bms3 algebra. It consists in the semi-direct sum of the vector fields on the circle equipped with the usual commutator acting on functions on the circle.

8 What would be interesting is to analyze in details what this classical central extension can teach us about quantum gravity in asymptotically flat 3d spacetimes. One should study whether a Cardy type formula works in this case also. Notice also that there are no black holes, but only conical defect solutions in this case. BMS3/CFT1 Charge algebra modes Y (θ) 1 copy of Wit algebra acting on i 1 iso(2, 1) charge algebra: relation to AdS 3 similar to contraction between so(2, 2) iso(2, 1) L ± m = 1 2 ( lp ±m ± J ±m ) l collaboration with G. Compère After Fourier analyzing, the algebra consists of 1 copy of the Wit algebra acting on the functions on the circle in a similar way than the Lorentz transformations act on the ordinary translations. Poincaré transformations in 3d are obtained by restricting to the subalgebra of generators with labels -1,0,1. In fact the bms3 algebra has been originally discussed in a paper by Ashtekar et al in What we have done though was the computation of the charge algebra. It turns out to contain a non trivial central extension between the two factors. A posteriori, it is clear that this is the only place for the central extension as it cannot appear in the one copy of the Wit algebra on account of the missing dimensional parameter l in the flat case. The relation to the AdS3 Virasoro case is by a contraction similar to the relation between so(2,2) and iso(2,1). More precisely, if one introduces a parameter of dimension length, there is an extension of the BMS algebra, and after redefining the generators, one finds both for the asymptotic symmetries and for the charges, including the central ones, the AdS3 results.

9 BMS3/CFT1 Solution space and conformal properties general solution parametrized by Θ = Θ(φ), Ξ = Ξ(φ) s uφ = e ϕ[ Ξ + u 0 ds 2 = s uu du 2 2dudr +2s uφ dudφ + r 2 e 2ϕ dφ 2, s uu = e 2ϕ[ Θ ( φ ϕ) φϕ ] 2r u ϕ, du e ϕ[ 1 2 φθ φ ϕ[θ ( φ ϕ) φϕ]+ 3 φϕ ]]. bms 3 transformation properties δ Y,T Θ = Y φ Θ +2 φ Y Θ 2 3 φy, δ Y,T Ξ = Y φ Ξ +2 φ Y Ξ T φθ + φ T Θ 3 φt, charges Q ξ [g ḡ, ḡ] 1 16πG K ξ1,ξ 2 = 1 8πG 2π 0 dφ 2π 0 dφ (ΘT +2ΞY ) [ ] φ Y 1 (T 2 + φt 2 2 ) φ Y 2 (T 1 + φt 2 1 ) More recently,with C.Troessaert, we have investigated in more details solution space as a preparation for the 4 dimensional case. It turns out that with the above ansatz, the general solution to the equations of motion can again be given, in fact just by solving ordinary differential equations. It is even easier than in the AdS3 case and involves, besides the conformal factor, 2 arbitray functions on the circle. The transformation properties reflect what we have found on the level of the charges, there is an inhomogeneous term, but not in the standard way. We have proven that the central extension is non trivial, i.e., cannot be absorbed by a redefinition of the generators, but what still needs to be done is a systematic analysis of the representations of the bms3 algebra.

10 BMS4/CFT2 Asymptotic flat spacetimes x 0 = u, x 1 = r, x 2 = θ,x 3 = φ, A, B, =2, 3 BMS ansatz ds 2 = e 2β V r du2 2e 2β dudr + g AB (dx A U A du)(dx B U B du) g AB dx A dx B = r 2 γ AB dx A dx B + O(r) Sachs: unit sphere γ AB = e 2ϕ 0γ AB 0 γ AB dx A dx B = dθ 2 + sin 2 θdφ 2 Riemann sphere ζ = e iφ cot θ 2, γ ABdx A dx B = e 2 eϕ dζd ζ determinant condition det g AB = r4 4 e4 eϕ fall-off conditions β = O(r 2 ), U A = O(r 2 ), V/r = 2r u ϕ + ϕ + O(r 1 ) After this long motivation and study of lower dimensional models, let me now turn to the main topic, which are 4d spacetimes that are asymptotically flat at null infinity. The ansatz by Bond, Metzner and Sachs in the early sixties for the line element is the same as before with one more angle and one more function U. Sachs chooses the leading order of the metric to be the one on the unit sphere, but the subsequent geometrical analysis by Penrose in terms of a conformal completion suggests to keep an arbitrary conformal factor. One then might as well go to the Riemann sphere and introduce the conformal factor with respect to the flat metric. In this definition of asymptotically flat spacetimes there is then also a condition which fixes the determinant of the 2-d metric. Without loss of generality, it can be chosen as follows. In addition there are fall off conditions.

11 BMS4/CFT2 Asymptotic symmetries asymptotic symmetries L ξ g rr =0, L ξ g ra =0, L ξ g AB g AB =0, L ξ g ur = O(r 2 ), L ξ g ua = O(1), L ξ g AB = O(r), L ξ g uu = O(r 1 ) general solution Y A = Y A (x B ) T = T (x B ) conformal Killing vectors of the sphere generators for supertranslations spacetime vector form faithful representation of bms 4 algebra [(Y 1,T 1 ), (Y 2,T 2 )] = (Ŷ, T ) Ŷ A = Y1 B B Y2 A Y1 B B Y2 A, T = Y1 A A T 2 Y2 A A T (T 1 A Y2 A T 2 A Y1 A ) standard GR choice: restrict to globally well-defined transformations SL(2, C) Y A generators of Lorentz algebra Again asymptotic symmetries are generated by vectors that leave this form of the metric, including the determinant condition, invariant. The general solution is parametrized by conformal Killing vectors of the unit or equivalently of the Riemann sphere and by an additional arbitrary function T which are the generators of so-called supertranslations. This result was basically obtained by Sachs in 1962, in fact on the unit sphere, but that does not matter so much. The question is then what are the conformal Killing vectors of the Riemann sphere. What people in GR have done is they have concentrated on those transformations that are globally well-defined. This singles out SL(2,C), or, in terms of generators, the Lorentz algebra. So the bms algebra is the semi-direct sum of the Lorentz algebra with the infinite-dimensional abelian ideal of supertranslations.

12 BMS4/CFT2 new proposal CFT choice : allow for meromorphic functions on the Riemann sphere solution to conformal Killing equation Y ζ = Y ζ (ζ), Y ζ = Y ζ( ζ) generators l n = ζ n+1 ζ, ln = ζ n+1 ζ, n Z T m,n = ζ m ζn, m, n Z commutation relations [l m,l n ] = (m n)l m+n, [ l m, l n ]=(m n) l m+n, [l m, l n ]=0, [l l,t m,n ] = ( l +1 2 m)t m+l,n, [ l l,t m,n ]=( l +1 2 n)t m,n+l. But in the context of quantum conformal field theories in 2 dimensions, people have faced the same question and chosen to allow for meromorphic functions on the Riemann sphere. The solution to the conformal Killing equation then involves an arbitrary function of zeta and one of zeta bar. In terms of the generators of the complexified Lie algebra, one then finds the following semi-direct extension of the Virasoro algebra. This algebra is to the Poincare algebra what the Virasoro algebra is to the Lorentz algebra.

13 BMS4/CFT2 solution space for simplicity ϕ =0 e = (2) g, h = g ζζ g ζ ζ + e, h = g ζ ζ g ζ ζ + e, Beltrami parametrisation g ζζ = BMS4 asymptotic solution parametrized by h = c r + d r 2 + O(r 3 ), 2eh 1 y, g ζ ζ = 2e h 1 y, g ζ ζ = e(1 + y) 1 y, g ζζ = 2 h e(1 y), g ζ ζ = 2h e(1 y), gζ ζ = 1+y e(1 y), M(u, ζ, ζ) c(u, ζ, ζ),n(u, ζ, ζ),d(ζ) β = 1 4r 2 c c 1 3r 3 (d c + dc)+o(r 4 ), U ζ = 2 c r 2 + N r 3 + O(r 4 ), V = 2M 1 [ c r r r c 2 + 3(c c + c c)+9 c c 1 2 [ u M = u 2 c + 2 c ] u c u c, ] ( N + N) + O(r 3 ), 3 u N = 4 M + 4( 2 c 3 c) + 3(c u c + c u c +5 u c c) u c c. real complex The analyis of solution space is work in progress. The results presented here are preliminary. For simplicity let us put the conformal factor to zero. Since the determinant of the 2d metric is fixed, it is also convenient to choose the Beltrami parametrisation. When making a power expansion in 1/r, the leading orders of the equations of motions then imply that asymptotically, solution space is characterized by a real function M and 3 complex functions, one of which depends only on 1 argument. Furthermore the time dependence on M and the complex function N is completely determined in terms of the leading order of the 2d metric.

14 BMS4/CFT2 transformation properties redefinitions M = M 2 c 2 c c + cċ, N = 4 (N 5 c c c c)+ 1 u M 3 transformations properties δ Y,T c = Y A A c +( 3 2 Y 1 2 δ Y,T d = Y d +2 Y d, δ Y,T ċ = Y A A ċ +2 Y ċ 3 Y + f c, Ȳ )c 2 2 f + fċ, δ Y,T M = Y A 3 A M + 2 ( Y + Ȳ ) M +4 f ċ + f( c c +2 2 ċ), δ Y,T N = Y A A N +( Y +2 Ȳ ) N T M T M f c ċ u (4 f ċ + f c c +2f 2 ċ). where f = T u AY A d primary field (2,0) c =0 Bondi s news ċ transforms like energy momentum tensor T =0 ċ =0 M c =0 primary field (3/2,3/2) N primary field (1,2) in progress... One can then work out the bms4 transformations of the functions parametrizing solution space. After some redefinitions, one finds the following transformations. From these, one begins to see several things. It follows for instance that d is a primary field of conformal dimensions (2,0). When the second derivative with respect to u vanishes, the variable c dot, also called Bondi s news transforms like the components of an energy momentum tensor, including the central, inhomogenous part. Also when T=0, and one concentrates on the Virasoro subalgebra and the news and its u derivative vanishes, \hat M and hat N are again primary fields of dimensions 3/2,3/2 and 1,2 respectively. This is very much work in progress. What we are working is the charge algebra, which is notoriously difficult at null infinity, a complete control on solution space and a better understanding of the representations of the new form of the bms4 algebra.

15 Appendix ADS3 general solution g AB dx A dx B = ( ) e 2ϕ r 2 +2 γ + r 2 e 2ϕ ( γ γ ++ γ ) dx + dx + + γ ++ (1 r 2 e 2ϕ γ + )(dx + ) 2 + γ (1 r 2 e 2ϕ γ + )(dx ) 2, γ ±± = l 2[ Ξ ±± (x ± )+ 2 ±ϕ ( ± ϕ) 2] γ + = l 2 + ϕ BMS3 symmetry vectors BMS4 symmetry vectors ξ u = f, ξ φ = Y + I, I = e 2ϕ φ f dr r 2 e 2β = 1 r r e 2ϕ φ f + O(r 2 ), ξ r = r [ φ ξ φ φ fu + ξ φ φ ϕ + f u ϕ ], u f = f u ϕ + Y φ ϕ + φ Y f = e ϕ[ T + u ξ u = f, ξ A = Y A + I A, I A = f,b dr (e 2β g AB ), r ξ r = 1 2 r( D A ξ A f,b U B +2f u ϕ), 0 du e ϕ ( φ Y + Y φ ϕ) ] f = f ϕ ψ f = eϕ[ T u 0 du e ϕ ψ ], ψ = D A Y A

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