CC BY
33
UDC 530.1; 539.1
Asymptotic symmetries at null infinity and local conformai properties of spin
coefficients
G. Barnich and P.-H. Lambert
Physique Théorique et Mathématique, Université Libre de Bruxelles and International Solvay institutes, Campus Plaine
C.P. 231, B-1050 Bruxelles, Belgium
E-mail: gbarnich@ulb.ac.be, Pierre-Henry.Lambert@ulb.ac.be
We show that the symmetry algebra of asymptotically flat four dimensional spacetimes at null infinity in the sense of Newman and Unti is isomorphic to the direct sum of the abelian algebra of infinitesimal conformai rescalings with bms4. We then work out the local conformai properties of the relevant Newman-Penrose coefficients, as well as the surface charges and their algebra.
Keywords: Gauge-gravity correspondence, Gauge symmetries, Asymptotic symmetries, Conformai symmetry, Central
extension, Surface charges, Newman-Penrose formalism, GHP formalism
1 Introduction
This conference proceedings summarizes the results of paper fl] to which we refer for detailed computations and discussions.
The definitions of asymptotically flat four dimensional space-times at null infinity by Bondi-Van der Burg-Metzner-Sachs [2, 3] (BMS) and Newman-Unti (NU) [4] in 1962 merely differ by the choice of the radial coordinate. Such a change of gauge should not affect the asymptotic symmetry algebra if, as we contend, this concept is to have a major physical significance. The problem of comparing the symmetry algebra in both cases is that, besides the difference in gauge, the very definitions of these algebras are not the same. Indeed, NU allow the leading part of the metric induced on Seri to undergo a conformal rescaling. When this generalization is considered in the BMS setting, it turns out that the symmetry algebra is the direct sum of the BMS algebra bms4 [5] with the abelian algebra of infinitesimal conformal rescalings [6], [7].
In this note we show that, as expected, the asymptotic symmetry algebra in the NU framework is again the direct sum of bms4 with the abelian algebra of infinitesimal conformal rescalings of the metric on Seri and thus coincides, as it should, with the generalized symmetry algebra in the BMS approach.
We then discuss the transformation properties of the Newman-Penrose coefficients parametrizing solution space in the NU approach, focussing on the in-homogeneous terms in the transformation laws that contain the information on the central extensions of the theory, and we finally study the associated surface charges and their algebra by following the analysis in the BMS gauge [8].
2 NU metric ansatz and asymptotic symmetries
The metric ansatz of NU can be written as
ds2 = Wdu2 -2drdu+gAB (dxA—VAdu)(dxB — VBdu),
(1)
with coordinates u, r, xA and where gAB dxAdxB = r2YABdxAdxB +rCABdxAdxB +o(r), with yaB conformally flat. Below, we will use standard stereographic coordinates Z = c°t 22e^, Z> ZABdxAdxB = e2(^dZdZ, i = p(u, x). There is also an additional condition, related to the fixing of the origin of the affine parameter of the null geodesic generators of the null hypersurfaces used to build the metric [4], which yields here
CA = o fi].
In the following we denote by DA the covariant derivative with respect to ZAB and by A the asso-
VA =
O(r-2^d W = — 2rd„<p + A <p + O(r-1), where Ai = 4e-2^dd<p with d = d^, d = d^.
The infinitesimal NU transformations are defined as those infinitesimal transformations that leave the form of the metric and the fall-off conditions invariant, up to a rescaling of the conformal factor 5ip(u, xA) = p(u, xA), and are in this case generated by
£“ = f,
eA = YA + IA, IA = —Ob f frCadr'gAB, (2)
T = —rduf + Z + J, J = dAf JT dr'VA,
with drf = 0 = drYA = drZ, Z = 1Af, duYA = 0, with YA a conformal Killing vector of ZAB, i.e. Yz =
Y = Y(Z), Yz = Y = Y(C) in the coordinates (Z, C)> and also with
_ ~ 1 ru
f = [t + 2yo du’e-^p, T = T(Z,Z), (3)
with p = DDa YA mid p = p — 2cT. Asymptotic Killing vectors thus depend on YA,T,iT and the metric, £ = £[Y, T, ¡T; g]. For such metric dependent vector fields, consider the suitably modified Lie bracket taking the metric dependence of the spacetime vectors into account, [£i,£2]m = [£1, £2] — ¿g£2 + ¿g2£1, where Jg £2 denotes the variation in £2 under the variation of the metric induced by £1; Jg gMV = gMV. Consider
now the extended bms4 algebra, i.e., the semi-direct sum of the algebra of conformal Killing vectors of the Riemann sphere with the abelian ideal of infinitesimal supertranslations, trivially extended by infinitesimal conformal rescalings of the conformally flat degenerate metric on Seri. The commutation relations are given
by [(Y1, T^ iT1), (Y2,T2,iT2)] = (y,T,iT) where YA = YB Bb Y2a — Y2b Bb Y14,
T = Y1AdA^2 — YAdAT1 + 2 (T^aYA — T2dAYA),
U3 = 0.
In these terms, one can show the following:
(4)
Theorem 2.1 The spacetime vectors £[Y, T, w; g] realize the extended, bms4 algebra in the modified Lie bracket,
£[Y1,T;1,W1; g],£[Y2,T2,w2; g]l = £Y,T,T; g], (5)
J M
in the bulk of an asymptotically flat spacetime in the sense of Newman and Unti.
Note in particular that for two different choices of the conformai factor <p which is held fixed, w = 0, the asymptotic symmetry algebras are isomorphic to bms4, which is thus a gauge invariant statement.
NU case instead [4], the free data characterizing solution space are described in terms of the spin coefficient a0 and its time derivative, and also in terms of the
(with a = 0,1, 2, 3, 4), five complex scalars representing all the components of the Weyl tensor. The explicit relations between the free data characterizing asymptotic solution space in both approaches were established for instance in [1].
Using the “eth” operators [9] defined for a field ns of spin weight s according to the conventions of [10] through Sns = P 1-sd(Psns), §ns = P1+sd(P-sns) with P = A/2e-^, where S, S raise respectively lower the spin weight by one unit and let Y = P-1Y and
Y = P-1Y. The conformai Killing equations and the conformai factor then become SY = 0 =_§Y and
0 = (SY + §Y)- Using the notation S = (Y, T, w), we have — Ss= 2w7AB for the background metric.
To work out the transformation properties of the NU coefficients characterizing asymptotic solution space, one needs to evaluate the subleading terms in the Lie derivative of the metric on-shell. This can also be done by translating the results from the BMS gauge, using the dictionary of [1], which yields in this case
1 =
-Ss &0 = [/d„ + YS + YS + - SY - - SY - - S2/
2
2
1
-Ss &0 = [/d„ + Y S + YS + 2SY - 2£]&0 - - S>,
-Ss *0 = [/d„ + Y S + YS +
5 - i
SY
- 305]*° + (4 - i)S/^0+1 :
(6)
with i = 1, 2, 3, 4.
3 Explicit relations between the NU and the BMS gauges and local conformal transformation laws of the NU coefficients
The choice of the radial coordinate in the definition of asymptotically flat space-times in the BMS
[2], [3], [5] and the NU [4] approaches differs but the relation between the two radial coordinates does not involve constant terms [1] and is of the form r' = r + O(r-1). This change of coordinates only affects lower order terms in the asymptotic expansion of the metric that play no role in the definition of asymptotic symmetries and explains a posteriori why the asymptotic symmetry algebras in both approaches are isomorphic.
In the BMS set-up, the general solution to Einstein’s field equations is parametrized by some functions [2], [3], [7] among which are the mass and angular momentum aspects, and the news tensors. In the
4 Surface charge algebra
In this section, it = 0 so that f = T + 2up and we use the notation s = (Y, Y, T) for elements of the symmetry algebra, which is given in these terms by [s1, s2] = Y where
Y = Y1SY2 - (1 ^ 2),
3> = Ÿ1SY2 - (1 ^ 2),
1
(7)
(YiS + YiS)T2 - 2^iT2 - (1 ^ 2).
The translation of the charges, the non-integrable piece due to the news and the central charges computed in [8]
gives here
f (*° + a°a °)+
+ Y (*? + ct°3<7° + - 3(CT°<7°))) + c.c.
es[^x, x] = 8“G/d2QV f ^°^° + c.c.],
KS1,S2 [X ] =
8nG 87tG , -
(4 /lS/2§Ä+
Qy ,°,° = -
-
8nG
*1 + <7°g<j ° + - g(a°a° )-
(8)
+ 1 cr0f192p2 — (1 ^ 2)) + c.c.
We recognize all the ingredients of the surface charges described in [11]. More precisely the angular (super-)momentum that we get is
(9)
and differs from Qnc given in equation (4) of [11] by the
u
has a similar structure to Penrose’s angular momentum as described in equations (11), (12), and (17a) of [11] in the sense that it also differs by a specific amount of linear supermomentum, but the amount is different and explicitly u-dependent, Qy,o,o = Qy7co0 + 1 uQo,o,gy .
The main result derived in [8] states that if one is allowed to integrate by parts, and if one defines the “Dirac bracket” through {QS1, QS2}*[X] =
—Js2QS1 [X] + 0S2 [—JS1X, X], then the charges define a representation of the bms4 algebra, up to a field dependent central extension, {QS1 ,QS2}* = Q[S1,S2] + KS1,S2, where Ks1,s2 satisfies the generalized cocycle condition K[S1,S2],S3 — ¿S3KS1,S2 + cyclic(1, 2, 3) = 0. This representation theorem can be verified directly in the present context [1].
To the best of our knowledge, except for the previous analysis in the BMS gauge, the above representation result does not exist elsewhere in the literature.
A major issue in these considerations is whether one uses the globally well-defined version of the bms4 algebra or a local version which contains the Virasoro algebra and involves an expansion in terms of Laurent series. The formulas presented above generally apply to both cases, except for divergences in the charges that appear in the second case and have to be handled properly. This is discussed in more details in [1].
Acknowledgements
The authors thank Cédric Troessaert for useful discussions. G.B. is Research Director of the Fund for Scientific Research-FNRS (Belgium). This work is supported in part by the Belgian Federal Science Policy Office through the Interuniversity Attraction Pole P6/11, by IISN-Belgium, by “Communauté française de Belgique - Actions de Recherche Concertées” and by Fondecyt Projects No. 1085322 and No. 1090753.
References
[1] Barnich G and Lambert P-H 2011 (Preprint 1102.0589v2)
[2] Bondi H, van der Burg M G and Metzner A W 1962 Proc. Roy. Soc. Lond. A 269 21
[3] Sachs R K 1962 Proc. Roy. Soc. Lond. A 270 103
[4] Newman E T and Unti TWJ 1962 Journal of Mathematical Physics 3 891-901 URL http://link.aip.org/link/7JMP/ 3/891/1
[5] Sachs R K 1962 Phys. Rev. 128 2851-2864
[6] Barnich G and Troessaert C 2010 Phys. Rev. Lett. 105 111103 (Preprint 0909.2617)
[7] Barnich G and Troessaert C 2010 JHEP 05 062 (Preprint 1001.1541)
[8] Barnich G and Troessaert C 2011 JHEP 1112 105 (Preprint 1106.0213)
[9] Newman E T and Penrose R 1966 J. Math. Phys. 7 863-870 URL http://link.aip.org/link/?JMP/7/863/l
[10] Penrose R and Rindler W 1986 Spinors and Space-Time, Volume 2: Spinor and Twistor Methods in Space-Time Geometry
(Cambridge University Press)
[11] Dray T and Streubel M 1984 Classical and Quantum Gravity 1 15-26 URL http://stacks.iop.org/0264-9381/1/15
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Received 01.10.2012
Г. Варниш, П.-Х. Ламберт
АСИМПТОТИЧЕСКИЕ СИММЕТРИИ ПРИ СВЕТОПОДОБНОЙ БЕСКОНЕЧНОСТИ И ЛОКАЛЬНЫЕ КОНФОРМНЫЕ СВОЙСТВА СПИНОВЫХ КОЭФФИЦИЕНТОВ
Показано, что алгебра симетрии асимптотически плоского четырёхмерного пространства-времени при свепоподобной бесконечности в смысле Ньюмана и Унти изоморфна прямой сумме абелевой алгебпы бесконечно малых конформных преобразований с bms4. Обсуждаются локальные конформные свойства соответствующих коэффициентов Ныомана-Пенроуза, а также поверхностные заряды и их алгебра.
Ключевые слова: конформная симметрия, коэффициенты, Ньюмана-Пенроуза.
Варниш, Г.
Свободный университет, Брюссель.
Campus Plaine С.P. 231, В-1050 Bruxelles, Бельгия.
E-mail: gbarnich@ulb.ac.be
Ламберт П.-Х.
Свободный университет, Брюссель.
Campus Plaine С.P. 231, В-1050 Bruxelles, Бельгия.
E-mail: gbarnich@ulb.ac.be
