Sachs’ free data in real connection variables

Journal of High Energy Physics, Nov 2017

We discuss the Hamiltonian dynamics of general relativity with real connection variables on a null foliation, and use the Newman-Penrose formalism to shed light on the geometric meaning of the various constraints. We identify the equivalent of Sachs’ constraint-free initial data as projections of connection components related to null rotations, i.e. the translational part of the ISO(2) group stabilising the internal null direction soldered to the hypersurface. A pair of second-class constraints reduces these connection components to the shear of a null geodesic congruence, thus establishing equivalence with the second-order formalism, which we show in details at the level of symplectic potentials. A special feature of the first-order formulation is that Sachs’ propagating equations for the shear, away from the initial hypersurface, are turned into tertiary constraints; their role is to preserve the relation between connection and shear under retarded time evolution. The conversion of wave-like propagating equations into constraints is possible thanks to an algebraic Bianchi identity; the same one that allows one to describe the radiative data at future null infinity in terms of a shear of a (non-geodesic) asymptotic null vector field in the physical spacetime. Finally, we compute the modification to the spin coefficients and the null congruence in the presence of torsion.

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Sachs’ free data in real connection variables

JHE Sachs' free data in real connection variables Elena De Paoli 0 1 2 3 Simone Speziale 0 1 2 4 0 CNRS , CPT, UMR 7332, 13288 Marseille , France 1 Piazzale A. Moro 2 , 00185 Roma , Italy 2 Via della Vasca Navale 84 , 00146 Roma , Italy 3 Dipartimento di Fisica, Universita di Roma La Sapienza 4 Aix Marseille Univ., Universite de Toulon We discuss the Hamiltonian dynamics of general relativity with real connection variables on a null foliation, and use the Newman-Penrose formalism to shed light on the geometric meaning of the various constraints. We identify the equivalent of Sachs' constraint-free initial data as projections of connection components related to null rotations, i.e. the translational part of the ISO(2) group stabilising the internal null direction soldered to the hypersurface. A pair of second-class constraints reduces these connection components to the shear of a null geodesic congruence, thus establishing equivalence with the second-order formalism, which we show in details at the level of symplectic potentials. A special feature of the rst-order formulation is that Sachs' propagating equations for the shear, away from the initial hypersurface, are turned into tertiary constraints; their role is to preserve the relation between connection and shear under retarded time evolution. The conversion of wave-like propagating equations into constraints is possible thanks to an algebraic Bianchi identity; the same one that allows one to describe the radiative data at future null in nity in terms of a shear of a (non-geodesic) asymptotic null vector eld in the physical spacetime. Finally, we compute the modi cation to the spin coe cients and the null congruence in the presence of torsion. aDipartimento di Fisica; Universita di Roma 3 1 Introduction 2 Sachs' free data and metric Hamiltonian structure 2.1 Bondi gauge and Sachs constraint-free initial data 2.2 Hamiltonian structure 3 Canonical structure in real connection variables 4.1 4.2 4.3 3.1 Tetrad and foliation 3.2 Constraint structure 4 Geometric interpretation Newman-Penrose tetrad The a ne null congruence 5 Bondi gauge 6 Conclusions A Spin coe cients B Congruence Torsionlessness of the a ne null congruence 4.4 Tertiary constraints as the propagating equations 5.1 Equivalence of symplectic potentials 5.2 Radiative data at future null in nity as shear `aligned' to I+ C Tetrad transformations and gauge xings D 2 + 2 foliations and NP tetrads D.1 Adapting a NP tetrad D.2 The Bondi gauge and Newman-Unti tetrad E Mappings to the -tetrad E.1 The Bondi gauge and Newman-Unti tetrad E.2 Areal r and Sachs' metric coe cients { 1 { Introduction Null foliations play an important role in general relativity. Among their special features, they admit a gauge- xing for which the Einstein's equations can be integrated hierarchically, and constraint-free initial data identi ed, as shown by Sachs [1]; and provide a framework for the description of gravitational radiation from isolated systems and of conserved charges, starting from the seminal work of Sachs, of Bondi, van der Burg and Metzner (henceforth BMS), Newman and Penrose (NP), Geroch and Ashtekar [2{12] (see also [13{15] and reference therein). These classic results are based on the Einstein-Hilbert action and the spacetime metric as fundamental variable, and provide a clear geometric scribed in terms of connections [10, 11].1 We then wish to provide a connection description of the physical degrees of freedom in the spacetime bulk, in the sense of constraint-free initial data for the rst-order action. Secondly, the connection description later led Ashtekar to the famous reformulation of the action principle of general relativity [ 19 ], which is at the root of loop quantum gravity. This approach to quantising general relativity suggests the use of connections as fundamental elds, instead of the metric. There exists a canonical quantisation scheme that leads to the well-known prediction of quantum discreteness of space [20]. This result uses space-like foliations, and the dynamical restriction to the quanta of space imposed by the Hamiltonian constraint are still not explicitly known, nonewithstanding constant progress in the eld. Quantising with analogue connection methods the constraint-free data on null foliations would allow us to study the quantum structure of the physical degrees of freedom directly.2 As a preliminary result in this direction, it was shown in [22] that at the kinematical level, discretisations of the 2d space-like metric have quantum area operators with a discrete spectrum given by the helicity quantum numbers. A stronger more recent result appeared in [23], based on covariant phase space methods and a spinorial boundary term, con rming the discrete area spectrum without a discretisation. What we would like is to extend these results within a Hamiltonian dynamical framework. The Hamiltonian dynamics of general relativity with real connection variables on a null foliation appeared in [24],3 and presents a few intricate structures, like the conversion of what Sachs called the propagating Einstein's equations into (tertiary) constraints. In this paper, we present three results. First, we use the Newman-Penrose formalism to clarify 1Another class of null hypersurfaces for which the connection description plays an important role is the one of isolated horizons [16{18]. 2For recent work towards the same goal but in metric variables, see [21]. 3For previous studies using complex self-dual connections see e.g. [25, 26]. { 2 { the geometric meaning of the various constraints present in the Hamiltonian structure studied in [24]. Second, we identify the connection equivalent of Sachs' free data as the `shear-like' components of an a ne4 null congruence; we show how they reduce to the shear of a null geodesic congruence in the absence of torsion, and how they are modi ed in the presence of torsion; we use the Bondi gauge to derive their Dirac brackets, and show the equivalence with the metric formalism at the level of symplectic potentials. Third, we explain the origin and the meaning of the tertiary constraints, and point out that the algebraic Bianchi identity responsible for the conversion of the propagating equations into constraints is the same one that allows the interpretation of the radiative data at future null in nity I + in terms of shear of a (non-geodesic) null vector eld `aligned' with I+. The identi cation of the dynamical part of the connection with null rotations (related on-shell to the shear) is a striking di erence with respect to the case of a space-like foliation, because these components form a group, albeit a non-compact one, unlike the dynamical components of the space-like formalism which are boosts (related on-shell to the extrinsic curvature). We have thus two senses in which a null foliation gives a simpler algebra: the rst-class part of the constraint algebra is a genuine Lie algebra (thanks to the fact that the Hamiltonian is second class), and the connection physical degrees of freedom form a group. The paper is organised as follows. We rst review useful background material on the Hamiltonian structure on null foliations: in section 2, with metric variables, including the use of Bondi coordinates and identi cation of constraint-free initial data and their symplectic potential; in section 3, with real connection variables. In section 4, we map the non-adapted tetrad used in the Hamiltonian analysis to a doubly-null tetrad, we identify the constraint-free data and study the e ect of the constraints on an a ne null congruence. We describe the modi cations induced by torsion in the case of fermions minimally and nonminimally coupled, as well as for a completely general torsion. We rederive the conversion of the propagating equations into constraints using the Newman-Penrose formalism, and single out one algebraic Bianchi identity responsible for it. In section 5 we specialise to Bondi coordinates, and discuss the Dirac bracket for the constraint-free data and the equivalence of the symplectic potential with the one in metric variables. We nally highlight that the same algebraic Bianchi identity relevant to the understanding of the tertiary constraints plays an interesting role for radiative data at I+. The conclusions in section 6 contain some perspectives on future work. We also provide an extensive appendix with technical material. This includes the detailed relation of our tetrad foliation to the 2 + 2 foliation used in the literature, of the metric coe cients we use to those of Sachs and of Newman and Penrose, the explicit expression of all NP spin coe cients in the rst-order variables, and some details on the mixing between internal boost gauge- xing and lapse xing via radial di eomorphisms. For the purposes of this paper, we will mostly restrict attention to local considerations on a single null hypersurface. We neglect in particular boundary conditions and surface terms. These carry of course very important physics, and we will come back to them in 4In the sense of being given by an a ne connection, a priori non-Levi-Civita, not of being a nely parametrised. { 3 { class of the two-dimensional induced metric along the initial slice, or alternatively its shear, plus corner data at the 2d space-like intersection. With some additional regularity assumptions, one can also use a 3 + 1 foliation by null cones radiated by a time-like worldline. See [27{31] for the formal analysis of solutions and existence theorems. Both evolution schemes are typically local because of the development of caustics, however for situations with su ciently weak gravitational radiation like those of [32], null cones can foliate all of spacetime. A case of special interest is the study of radiating isolated gravitational systems in asymptotically at spacetimes. In the asymptotic 2 + 2 problem, one puts the second null hypersurface at future null in nity I+, and the foliation describes null hypersurfaces (or null cones) attached to I+. In this case the assignment of initial data is subtler (see e.g. [33]), because of the compacti cation involved in the de nition of I. In particular, I+ is shear-free by construction. Nonetheless, the data are still described by an asymptotic shear, transverse to I + [5, 7, 10], and Ashtekar's result was to show that these degrees of freedom and the phase space they describe are better thought of in terms of connections living on I+, a construction which is useful for the understanding of conserved charges. Notice that one can not take I `hole' at i+ where tails and bound states escape null in nity (see e.g. [34]), nor I the same reason. We will mostly focuse on local properties of null hypersurfaces, and for not discuss the non-trivial features associated for instance with boundary data at corners, residual di eomorphisms, caustics and cone-vertex regularity, for which we refer the reader + itself as null cone of a 3 + 1 foliation, because of the to literature cited above and below. 2.1 Bondi gauge and Sachs constraint-free initial data The Bondi coordinate gauge is speci ed as follows: we take spherical coordinates in a local patch of spacetime, x = (u; r; ; ), with the level sets of u to provide a foliation into null hypersurfaces . du is thus a null 1-form, implying the gauge- xing condition g00 = 0, and the associated future-pointing null vector l = g @ u is tangent to the null geodesics of { 4 { HJEP1(207)5 Bondi coordinates: (u; r; θ; φ) r =const u =const S0; (θ; φ) world-line. as follows, g 0 = B intersect on a space-like 2d surface S0. When the two null hypersurfaces are intersecting light cones, as in the picture, S0 has topology of a sphere. The (partial) Bondi gauge is such that ( ; ) are value of r, unless the spacetime has special isometries. Right: further requiring suitable regularity conditions one can consider also a local 3 + 1 foliation of light-cones generated by a time-like HJEP1(207)5 . The second gauge condition is to require the angular coordinates xA = ( ; ), A = 2; 3, to be preserved along r, i.e. l @ xA = 0. This implies g0A = 0 and makes r a parameter along the null geodesics: the level sets of r thus provide a 2 + 1 foliation orthogonal to the null geodesics. At this point, the metric and its inverse can be conveniently parametrized e 2 Vr + ABU AU B e 2 0 ABU B1 0 AB C ; A g 0 0 = B in terms of seven functions ( ; V; U A; AB). Being the coordinates adapted to the 2 + 2 foliation de ned by u and r, gAB AB is the metric induced on the 2d space-like surfaces, and we denote its determinant and its inverse AB. The gauge- xing has the property that gAB = AB, so it is analogue to the shift-free (partial) gauge N a = 0 for a space-like foliation. There still remains one coordinate freedom, for which two di erent choices are customary in the literature: we can require as in [1, 3, 4] the radial coordinate to be an areal parameter R (called `luminosity distance' by Sachs), namely we can follow the Newman-Penrose (NP) literature [5, 6] and require g01 = 1, with no restrictions on , which makes r an a ne parameter for the congruence generated by l . plays the role of the lapse function in the canonical theory, and these two choices correspond to two di erent gauge- xings of the radial di eomorphism constraint. Accordingly, we will denote from now on e2 = N > 0. In the following, we will often keep this last gauge xing unspeci ed, for our results to be easily adapted to both choices. We will then refer to the x p = R2f ( ; ); or partial gauge- xing g00 = 0 = g0A as partial Bondi gauge.5 5A third option to complete the partial Bondi gauge is to take dr null, so to have also g11 = 0. This choice, used in the original Sachs paper [1], is not adapted to the asymptotic problem, and will be not considered in the following. { 5 { To set up the characteristic 2 + 2 initial-value problem, one chooses initial data on two null hypersurfaces intersecting on a space-like 2d surface S0, see gure 1. Working with a null foliation, any xed value of u identi es the rst null hypersurface. On the other hand, with r a ne or areal at most one r =constant hypersurface will also be null, for a generic spacetime. Its location can be xed with a measure-zero gauge- xing g11jr0 = 0. Then, as shown originally in [1] (see also [13, 35, 36]), constraint-free initial data for general relativity can be identi ed with the conformal class of 2d space-like metrics AB, of which we take the uni-modular representative data at the corner S0 between the two initial slices.6 AB := 1=2 AB; supplemented by boundary Up to the measure-zero corner data, the two independent components of AB are the two physical degrees of freedom of general relativity on a null hypersurface. In the associated hierarchical integration scheme, the Hamiltonian constraint can be solved as a radial linear equation for V , and one can identify the propagating equations for the constraint-free data as (the traceless part of) the projection of the Einstein's equations on the space-like surface. These give the evolution of AB away from the initial slice. The price to pay for the identi cation of constraint-free data is that the dynamical spacetime can be reconstructed only locally in a neighbourhood of the characteristic surface (neighbourhood that may well be smaller then the maximal Cauchy development, see e.g. [27]), as caustics develop and stop the validity of the coordinate patch. See e.g. [13, 29, 37, 38] for various discussions on this. The geometric interpretation of the constraint-free data is most commonly given in terms of the shear of null geodesic congruences, which is directly determined by the induced 2d metric. To see this, let us consider the normal 1-form l = it is automatically geodesic and twist-free; and since the level sets of u provide a null foliation, it is a nely parametrised. The associated congruence tensor coincides then with the Lie derivative of the induced metric, which in partial Bondi gauge is proportional to the radial derivative, rAlB = 1 2 $l AB = 1 2N This surface tensor can be familiarly decomposed into shear AB and expansion as the trace-less and trace parts, 1 2 $l AB = p 2 $l AB + 1 2 AB$l ln p = AB + 1 2 AB : (2.2) (2.3) Hence, the shear of the null congruence carries the same information of the conformal 2d metric, up to zero modes lost in the derivative and which are part of the corner data. The fact that (the bulk of the) constraint-free data can be described in terms of shear will allow us to easily identify them in the rst-order formalism, where r is an a ne connection. 6Explicitly, Sachs' also xes the residual hypersurface gauge, and provides the corner data They provide the area of S0, the initial expansion of the null geodesic congruences along the two hypersurfaces, and the non-integrability of the two null directions: UA;1 gives in these coordinates the Lie bracket { 6 { Here we used the Bondi gauge in order to identify the tangent vector eld to the null geodesic congruence with a coordinate vector, thus simplifying Lie derivatives. A 2d space-like metric in , its Lie derivative de ning a shear, and associated Sachs' propagating equations, can be identi ed without this gauge- xing: it su ces to use a 2 + 2 decomposition, either in terms of two scalar elds de ning a 2 + 2 foliation (one being u), or in terms of a null dyad (one element being l ), as we will review below. The role of the gauge- xing is nonetheless crucial to specify the explicit integration scheme of the constraints and the other eld equations. Hence, it is possible to talk about physical degrees of freedom in a completely covariant way, as often done in the literature, although only once the gauge is completely xed one can truly identify constraint-free initial data. 2.2 Hamiltonian structure The fact that the constraint-free data can be either described by the metric or the shear, its null-radial derivative, captures a well-known property of eld theories on the light cone: the momentum conjugated to the elds does not depend on velocities, but on the null radial derivative of the eld. Consider for instance a scalar eld in Minkowski spacetime. De ning x := t momentum is r, and choosing x+ as `time' for the canonical analysis, the conjugate : where A = 2; 3 are the transverse coordinates. The independence of the momentum from the velocities gives rise to a primary constraint := , which is second class with itself, up to zero modes, see e.g. [24]. In the following, we will refer to this constraint as light-cone condition. This fact, which is just a direct consequence of the fact that the normal vector to a null hypersurface is tangent to it, means that the momentum is not an independent variable, and can then be eliminated from the phase space. The physical phase space has thus 1 1 dimensions per degree of freedom, instead of 1 2 as in the spacelike formulation, and the elds satisfy Dirac brackets de ned by a suitable regularisation of @ 1. Since we are not interested in this paper in the subtle infrared issues and boundary conditions, let us content ourselves to describe the symplectic structure of the theory looking at the symplectic potential. To that end, one can use the covariant phase space method (see e.g. [39]), and read the symplectic potential from the variation of the action in presence of a null boundary. Consider for simplicity a free scalar eld, and a null boundary given by a single light-cone ruled by x . Then the variation of the action gives the following boundary contribution, (2.4) (2.5) This symplectic potential shows that the conjugate momentum to satis es the light-cone condition (2.4), and announces the presence of @ 1 in the Dirac bracket among the 's. The same structure arises in gauge theories (see e.g. [40]) and linearised general relativity around Minkowski [41, 42]: the physical phase space has 1 physical degree of freedom (a transverse mode in these examples), and the conjugate momentum is given by the null radial derivative of the mode itself. Remarkably, it is also true 1 dimensions for each in full, non-linear general relativity, with the momentum given by the shear, again a null radial derivative of the physical degrees of freedom as shown in (2.2). The Hamiltonian analysis of general relativity on a null hypersurface has been performed in [36] using the 2 + 2 formalism of [35]. Starting with a covariant kinematical phase space of canonical variables (g ; := L= @ug ), one nds 6 rst class and 6 second class constraints, for a resulting 2-dimensional physical phase space, as expected. The six rst class constraints split in 3 hypersurface di eomorphism generators plus three primary constraints imposing the vanishing of the conjugate momenta to the chosen shift vectors. The six second class are: the null hypersurface condition g00 = 0, which in turns gauge- xes the Hamiltonian constraint and makes it second class;7 two light-cone conditions, the non-linear version of (2.4); the vanishing of the momentum conjugated to the lapse N , and the vanishing The analysis of [36] is general and does not require the Bondi gauge: we introduce a 2 + 2 foliation by two closed 1-forms, n = d locally, with = 0; 1, normals to a pair of hypersurfaces. Instead of lapse and shift, we have two shift vectors and a `lapse matrix' is to take a null foliation de ned say by the level sets of 0, so that N 00 = 0 = N11, and the lapse (i.e. the Lagrange multiplier of the Hamiltonian constraint) turns out to be the o -diagonal component, N01 = N .9 The induced space-like metric on the 2dimensional surface orthogonal to both normals is then = g N n n . In this formalism, we can identify covariantly the two physical degrees of freedom with ; their propagating equations as the two components of the Einstein equations obtained from the trace-less projection onto the 2d surface; and their Hamiltonian counterpart as the multiplier equations arising from the stabilisation of the two light-cone conditions. If we adapt the null coordinate, 0 = u, we have n1 = N01n0 = N l . Unlike l , n1 has non-vanishing a nity, given by k(n1) = $n1 ln N , and its shear and expansion are N times those of l . The partial Bondi gauge corresponds to putting to zero one of the two shift vectors, and only in this gauge the coordinate vector @ 1 is tangent to the null geodesics on . As discussed above, the gauge- xing is convenient for many reasons, principally to provide the explicit integration scheme of the Einstein's equations, in particular solving the constraints. Another advantage is that due to the presence of complicated second class constraints, it is di cult to write the explicit Dirac bracket for the physical phase 7Up to zero modes: measure-zero `parallel' time di eomorphisms are still allowed. For instance, these contain the BMS super-translations [4] for asymptotically at spacetimes. 8This last constraint may look puzzling. The problem is that imposing g00 strongly in the action would lead to a variational principle missing one of the Einstein's equations. To avoid this `missing equation', the Hamiltonian in [36] is rst constructed with arbitrary g00, and g00 = 0 is later imposed as initial-value constraint on the phase space. The additional constraint @ug00 = 0 then simply arises as a secondary constraint preserving the rst one under evolution. As explained in [24], an advantage of working with a rst order formalism is that one does not need this somewhat arti cial construction: we can impose the gauge- xing condition strongly in the action and still have a complete well-de ned variational principle, thanks to the appearance in the action of the variable canonically conjugated to g00. Furthermore, the on-shell value of the Lagrange multiplier for g00 = 0, which is xed by hand in [36], comes up dynamically as a multiplier equation. 9The sign we use in this de nition is opposite to the one of [36], to match with our earlier choice N > 0. { 8 { space. Gauge- xing gets rid of gauge quantities and simpli es this problem. It becomes for instance straightforward to write the symplectic potential purely in terms of physical data. For our purposes, we specialise here the analysis of [36] to the partial Bondi gauge, adapting coordinates so that 0 = u and requiring g0A = 0, but keeping r un xed as to see explicitly the role of lapse and p . This partial gauge- xing eliminates various gauge elds from the phase space, and one can isolate the induced 2d metric AB and its conjugate momentum density, which turns out to be ^ AB := p AB = L _ AB = p 2 p ( AB CD AC BD)$n1 AB $n1 ln N + 1 2N CD $n0 N 00 ; (2.6) (2.7) (2.8) in terms of the dual basis (n0; n1) de ned above. Taking the trace-less and trace parts, it is immediate to identify them as the shear and expansion of the null-geodesic congruence of n1, AB := 1 2 AB AB = p 2 $n1 AB = (n1)AB; AB = (n1) 2k(n1) 1 N $n0 N 00: The rst equation above is precisely the light-cone condition (2.4) for non-linear gravity: the two physical momenta are the null radial derivatives of the two physical modes of the metric, namely, the shear of n1. The second equation shows that the trace of the momentum does not carry any additional information, although this may require a few words: rst, the expansion can be determined from the dynamical elds (up to boundary values) using the Raychaudhuri equation; the lapse can always be xed to 1 with a radial di eomorphism as mentioned above, thus removing the non-a nity term;10 nally, the last term vanishes using the equations of motion. In this partial Bondi gauge, the symplectic potential computed in [36] reads11 = Z d3x ^ AB AB = d3x (n1)AB ^AB + ( (n1) + 2k(n1)) p ; (2.9) Z p 2 where we used p AB. Notice also that the shear term can be rewritten using AB ^AB = p AB AB. AB = AC BD BD and de ned the densitised inverse metric ^AB := The non-a nity term vanishes if we x a constant lapse, and using the explicit metric form of shear and expansion, the symplectic potential takes the form = Z d3x $n1 AB ^AB + $n1 ln p p : (2.10) 10Canonically, the fact that changing r can be used to x N = 1 follows from the fact that lapse transforms under radial di eos like the radial component of a tangent vector. The alternative gauge- xing, r areal coordinate with lapse free, turns the non-a nity term into a corner contribution to the symplectic potential, see e.g. [38]. As mentioned above, we do not discuss corner terms in the present paper. 11As usual, deriving the symplectic potential requires an integration by part. Although [36] does not give the associated boundary term, this is known to be 2 R ( + k)p , see e.g. [43]. Note the di erent factors of 2 between the boundary term and the symplectic potential. { 9 { The rst term has precisely the form (2.5) for the 2 physical degrees of freedom, which is the main point we wanted to make. The second term is just a corner contribution thanks to the Bondi gauge. In this paper we are interested in bulk degrees of freedom, hence we neglect corner terms in the symplectic potential. This symplectic potential for the shear, here adapted from [36] to the Bondi gauge, can also be derived with covariant phase space methods (see e.g. [39]), without referring to a special coordinate system but only to the eld equations. It plays a crucial role in the study of BMS charges at null in nity (see e.g. [12, 14, 15]), which has recently received much attention for its possible relation to the information black hole paradox argued for in [44]. For a careful treatment of caustics, corner data and residual di eos, see [38, 45], as well as [46] in a related context. For a more general expression of without a full foliation and a discussion of corner terms without any coordinate gauge xing, and its relevance to capture the full information about the charges, see [47]. See also [43, 48{50] for additional discussions on corner terms. 3 3.1 Cartan action Canonical structure in real connection variables Tetrad and foliation In this section we brie y review the canonical structure of general relativity in connection variables on a null hypersurface [24]. In units 16 G = 1, we work with the EinsteinS[e; !] = where eI is the tetrad 1-form, and F IJ (!) = d!IJ + !I K ^ !KJ the curvature of the spin connection !IJ . As in the ADM formalism, we x a 3+1 foliation with adapted coordinates x = (t; xa), and hypersurfaces as follows [51{53], described by the level sets of t. We parametrise the tetrad e0 = N^ dt + iEaidxa; ei = N aEaidt + Eaidxa: The hypersurface normal is then the soldering of the internal direction xI+ := (1; i): n := eI x+I = ( N; 0; 0; 0); N = N^ N aEai i: (3.1) (3.2) (3.3) For space-like , the usual tetrad adapted to the ADM coordinates is recovered for vanishing i, which makes e0 parallel to the hypersurface normal. Using a non-adapted tetrad may appear as an unnecessary complication, but has the advantage that allows one to control the nature of the foliation. The metric induced by (3.2) on is qab := eIaebJ IJ = Xij EaiEbj ; Xij := ij i j ; det qab = E2(1 2); (3.4) where 2 := i i. It is respectively space-like for 2 < 1, null for 2 = 1, and time-like for 2 > 1. In other words, we control with i the signature of the hypersurface normal, while e0 is always time-like. We are interested here in the case of a foliation by null hypersurfaces. Notice that even though the induced hypersurface metric is degenerate, we can still assume an invertible triad, with inverse denoted by Ea. This means that we can use the triad determinant, E := i det Eai 6= 0, to de ne tensor densities. We denote such densities with a tilde respectively above or below the tensor, e.g. Eia := EEia for density weight 1 and Eai := E 1Eai for density weight 1. The triad invertibility is an advantage of the tetrad formalism for null foliations, and it allows us to write the null direction of the induced metric on as (Eia i)@a. Further, although the induced metric qab is not invertible, we can raise and j lower its indices with the triad. We de ne the projector qab := EaiEb Xij , which projects qab := EiaEjbX ij , which satis es qabqbc = qac. On the other hand, N^ and N a should not be immediately identi ed with the lapse and shift functions, de ned as the Lagrange multipliers of the di eomorphism constraints. The true lapse can be identi ed from (3.3) or by computing the tetrad determinant, which turns out to be e = N E. As for the shift vector, there is no canonical choice on a null foliation, corresponding to the fact that there is no canonical Hamiltonian.12 Following the canonical analysis of [24], to be recalled below, we keep N a as the shift vector. In terms of the lapse N , the metric associated with the tetrad (3.2) reads with inverse the tetrad, We then have and g = N 2 + N aN bqab 2N N aEai i qbcN c qacN c N Eai i g = 1 N 0 Ea i i Eb i i N EiaEib + (N aEib + N bEa) i i qab N Ebi i ! ; ! : The coordinate t being adapted to the null foliation, gab qab is the degenerate induced metric on . We can also write the projector on the 2d space-like spaces in a covariant form, using the null dyad provided by the internal null vectors xI = ( 1; i) soldered by xI := ( 1; i); x = eI x I = ; x+ x = 2: (3.7) ( ( N; 0) (N + 2N aEa ; 2Ea ) ? := 1 2 x+x 1 2 x x+ = 0 qabN b qab 0 ! ; := g x+( x ) = 12In the sense that it is not possible to express the Hamiltonian constraint purely in terms of hypersurface data, see for instance [24] and [36]. (3.5) (3.6) (3.8) (3.9) is the induced metric in covariant form. For later purposes, let us identify here the propagating Einstein's equations, which are given by the components (? GT)ab := a ? ( ? b ) 1 is the traceless part of the projector on S for symmetric hypersurface tensors, and we used explicit form of (3.10) is given in [24], and it will not be needed here. An advantage of the tetrad formulation is that we can perform the canonical analysis with the 3 + 1 null foliation [24], without the need of introducing a further 2 + 2 foliation like in the metric case. Nonetheless, it is instructive to review how the two formalisms compare in the absence of torsion. Our coordinates are adapted to the 3 + 1 foliation by null hypersurfaces with normal 1-form dt, and to match notations with the literature, we rename from now on t = u; however the 2d space-like spaces de ned by (3.8) are in general not integrable, hence they do not foliate spacetime. Nonetheless, we can choose a 2 + 2 foliation and adapt our tetrad to it. For the sake of simplicity let us choose the foliation given by the normals n0 = du; n1 = dr; (3.12) so that our coordinates xa = (r; xA) are already adapted, and the induced 2d metric is AB gAB = qAB. To adapt the null dyad x to this foliation we use the translational part of the ISO(2) group stabilising xI+ to remove the components x A = EAi i = 0. This gauge transformation makes the tangent vectors to fSg integrable. The same can be done in the Newman-Penrose formalism, see appendix D for details and a general discussion. Comparing then the metric coe cients of (2.1) and (3.6) we see that the lapse functions used in the metric and connection formulations di er by a factor Er i. This can be always i set to one with an internal boost along xI+, as explained in the next section. Hence, using this boost and the translational part of the stabiliser we can always reach the internal `radial gauge' E = p g where the equivalence follows from the invertibility of the triad. In this internal gauge N coincides with the lapse of the metric formalism, given by 1=g01 in adapted coordinates, and p g = N E = N p , and the induced metrics coincide, g x+( x ) = n n . Proofs and more details on the relation between the -tetrad and the 2+2 Eai i = (1; 0; 0) , E ir i = 1; X ij Ejr = 0; (3.13) formalism are reported in appendix E. 3.2 Constraint structure On a null hypersurface, each degree of freedom is characterised by a single dimension in phase space, as recalled above. This means that the constraint structure associated to S = where the gravitational action should lead to a phase space of dimensions 2 eventual zero modes and corner data, not discussed here). We now review from [24] how this counting comes about, as the result has some peculiar aspects that we wish to analyse in this paper. From (3.1), we see that the canonical momentum conjugated to !IJ is PIaJ := a (1=2) abc IJKLebK eL, namely, it is simple as a bi-vector in the internal indices. This rec sults in a set of (primary) simplicity constraints, which xing an internal null direction, can be written in linear form as Ia := IJ KLPKa LxJ+ = 0. Two di erent canonical analysis were presented in [24]. The rst is manifestly covariant, with only 2 = 1 as a gauge- xing condition. The second gauge- xes instead all three components, that is i = ^i for a xed ^i with ^2 = 1. Since in this paper we are interested in the identi cation of constraint-free data that arises through a complete gauge- xing, we recall only the details of the second analysis, and refer the reader interested in the covariant expressions to [24]. Working with a gauge- xed internal direction, we can solve explicitly the primary simplicity constraints in terms of P0ai = Ea; Piaj = 2E[ai j]. The kinetic term of the action is i then diagonalised by the same change of connection variables as in the space-like case [52], !a0i = ai !aij j ; !aij = ijk r~kl + 1 2 klm!~m l Ea; (3.14) with r~ij symmetric. After this change of variables and an integration by parts, the ac2(Eia@t ai + ij @tr~ij + i@t!~i) + ij ij + i'i + nIJ GIJ + N aDa + N H; (3.15) GIJ := DaPIaJ ; Da := PIbJ FaIbJ + !aIJ GIJ ; H := EiaEjbFaijb are the gauge and di eomorphism constraints, written in covariant form for practical reasons. Notice that as in the space-like case, the generator of spatial di eomorphism includes internal gauge transformations (and accordingly, we have nIJ = !IJ 0 N a!aIJ ). Next, the constraint imposes the vanishing of the momentum conjugated to rij , and is the left-over of the primary simplicity constraints in this non-covariant analysis. Finally, the constraint ij = ij 'i = i ^ i (3.17) (3.18) gauge- xes the internal vector. In particular, the projection ( i + ^i)'i gives the nullfoliation condition 2 = 1, namely g00 = 0, and its stabilisation plays an important role in recovering all of Einstein's equations.14 13In [24] we rescaled the action by a factor 1=2, to avoid a number of factors of 2 when computing Poisson brackets. Here we restore the conventional units. Accordingly, the parametrization of PIaJ in terms of Eia, as well as the explicit expressions for the constraints presented below in (3.16), are twice those of [24]. 14This plays the role of the @ug00 = 0 condition of [36], and the advantage of the rst-order formalism is that it can be imposed prior to computing the Hamiltonian. HJEP1(207)5 The phase space of the theory has initially 36 dimensions, with Poisson brackets f ai(x); Ejb(x0)g = fr~ij (x); kl(x0)g = The explicit form of the constraints is considerably more compact and elegant than in the metric case [36], a fact familiar from the use of Ashtekar variables in other foliations. On the other hand, many of the constraints are second class. The reader familiar with the Hamiltonian analysis in the space-like case will recall that the stabilisation of the primary HJEP1(207)5 simplicity constraints leads to six secondary constraints which are second class with the primary. The secondary constraints thus obtained, together with the six Gauss constraints, recover half of the torsion-less conditions; the remaining half goes in Hamiltonian equations of motion. In the null case the situation becomes more subtle: there are again six secondary where ij = (iklEkaEbj)@aElb + (iklEka l a j) M ij;klrkl; M ij;kl = (ikm j)ln Xmn: These have the same geometric interpretation of being six of the torsion-less conditions. However, only four of them are now automatically preserved. This is a consequence of the fact that (3.21) has a two-dimensional kernel: ij klM kl;mn 0, where ikjl is the internal version of the symmetric-traceless projector (3.11) obtained via the triad. Then, stabilisation of the two secondary constraints ^ ij = kl ij kl; requires two additional, tertiary constraints ab := 1 2 cadbEi(c d)ef Fe0fi j Feifj = 0: As pointed out in [24], the two constraints (3.22) are the light-cone conditions imposing the proportionality of physical momenta to the hypersurface derivatives in the null direction: as we will show below, they reproduce precisely the metric relation (2.7) between momenta and shear. What is peculiar to the formalism, is that this condition is not automatically preserved under the evolution, but requires the additional constraints (3.23). These additional constraints are not torsion-less conditions; they will be discussed in details in section 4.4 below. Concerning the nature of the constraints and the dimension of the reduced phase space, we have the following situation. The hypersurface di eos Da are rst class, but not the Hamiltonian H, which forms a second class pair with i'i. The other two components Xij 'j gauge- x two of the six Gauss constraints, those that would change the internal direction. The other four Gauss constraints remain rst class. This is di erent from the canonical analysis on a space-like or time-like hypersurface, where xing the internal direction gives a 3-dimensional isometry group. Here instead we have a 4-dimensional isometry group, given by the little group ISO(2) of the internal direction given by i, plus boosts along i . The fact that the isometry group on a null hypersurface is one dimension larger than for other foliations is of course a well-known property, that led Dirac himself to suggest the use of null foliations as preferred ones. In the context of rst-order general relativity with complex self-dual variables, it has for instance been pointed out in [25, 26]. However, there is a subtle way in which this extra isometry is realised in our context, because the action of internal boosts along i mixes with that of radial di eomorphisms. Let us spend a few words explaining it. Notice that right from the start we xed to unity the 0-th component of the internal null direction xI+. This choice, implicit in the parametrization (3.2) of the tetrad, deprives us of the possibility of changing i with an internal boost along i, since in the absence of a variable x0+ this would not preserve the light-likeness of the internal direction. Nonetheless, the explicit calculation of the constraint structure shows that K := G0i i is still a rst class constraint: simply, its action is not to change i, which it leaves invariant, but rather to rescale the lapse function. Using the transformation properties for Lagrange multipliers (see e.g. [54]), we nd for the smeared constraint the transformation K ( ) . N = e N: (3.24) In other words, the lapse function is in our formalism soldered to the extent of the internal null direction, see (3.7), and this is the reason why it transforms under internal radial boosts. As already discussed at the end of section 3.1, our lapse coincides with the lapse of the metric formalism only if we x the radial boosts to have Er i = 1. Hence, there i is in our formalism a partial mixing of the action of internal boosts along i and radial di eomorphisms. To complete the review of the constraints structure, it remains to discuss the simplicity constraints. They are all second class, but in di erent ways: ^ ij among themselves, just like those encoding the light-cone conditions (2.4), the remaining four class with four of the primary ij , and the remaining two ij are second class with the two tertiary constraints. The overall canonical structure established in [24] leads to the following diagram, where the arrows indicate which constraints are mutually second class: ij are second HJEP1(207)5 (3.25) We have 7 rst class constraints (forming a proper Lie algebra), and 20 second class constraints, for a 2 13-dimensional physical phase space, as expected for the use of a null hypersurface. Among those, the pair Hamiltonian-null hypersurface condition. 4.1 Geometric interpretation To elucidate the geometric content of the canonical structure in the rst order formalism, it is convenient to use the Newman-Penrose (NP) formalism. To that end, we want to map our tetrad (3.2) to a doubly-null tetrad (l ; n ; m ; m ), where l n = We have already partially done so, when we introduced the soldered internal null vectors x = e xI ; x I = ( 1; i), which provide the rst pair. For the second pair, we have to choose a spatial dyad for the induced metric (3.9), that is = 2m( m ); we can do so taking m to be a complex linear combination of the two orthogonal tetrad directions X ij ej , normalised by m m = 1. The set so de ned is an NP tetrad. Notice that x+ = to the null hypersurface. The minus sign in front of the second vector is to follow the conventions to have all vectors future-pointing. Before adopting the traditional notation with l and n for the rst two vectors, let us brie y discuss the frame freedom. Using the nomenclature of [55], we have rotations of class I leaving l unchanged, of class II leaving n unchanged, and of class III rescaling l and n and rotating m : l 7! A 1l ; n 7! An ; m 7! ei m : Conforming with standard literature on null hypersurfaces, we want the rst null co-vector to be normal to the null hypersurface and future pointing, that is l / its `normalisation', a reasonable choice is to take it proportional to the lapse function, like in the space-like Arnowitt-Deser-Misner (ADM) canonical analysis: lADM = analogy with ADM is con rmed by Torre's analysis, which as we recalled above, identi es in n1 lADM the normal relevant to the Hamiltonian structure, namely whose shear gives the conjugate momentum in the action. However, most of the literature on null hypersurfaces uses a gradient normal, l = @ u, and we'll conform to that, by taking 1 N l = x+; n = N 2 x : This rescaling of x means paying o a large number of N factors in the spin coe cients, see the explicit expressions reported in appendix A. In any case, the relation between the two choices is a class III transformation, and all NP quantities are related by simple and already tabulated transformations that can be found in [55], some of which are reported in appendix A.15 15The rescaling also means that while all Lorentz transformations of (4.2) are generated canonically via GIJ , this is not the case for (l; n) de ned via (4.4): we disconnect the canonical action of the radial boost K , which leaves them invariant instead of generating the class III rescaling. We see then again that lADM = x+ is a more canonical choice of null tetrad adapted to the foliation. (4.1) (4.2) (4.3) (4.4) We x from now on the following internal direction, and introduce the notation v iv3) for the internal indices M = 2; 3 orthogonal to it. This choice is done only for the convenience of writing explicitly the tetrad components of m and m when needed, and we will keep referring to i in the formulas as to make them immediately adaptable to other equivalent choices. Summarising, our NP tetrad and co-tetrad, and their expressions in terms of the metric coe cients (3.2), are (4.5) (4.6a) (4.6b) (4.6c) (4.7a) (4.7b) (4.7c) and 1 N N 2 1 = p (e2 2 l = n = m 1 N (e0 + e1 ) = 0; Ea i ; i (e0 e1 ) = 1 N 1; N a ie3 ) = (0; Ea ); 1 2 N Eia i ; l = n = m 1 = p (e2 N 2 2 ( e0 + e1 ) = ( 1; 0); (e0 + e1 ) = N 2 ie3 ) = (N aEa ; Ea ): (N + 2N aEai i); N Eai i ; The NP tetrad thus constructed is adapted to a null foliation like the one used in most literature [56, 57, 64]. A detailed comparison and discussion of the special cases corresponding to a tetrad further adapted to a 2 + 2 foliation or to the Bondi gauge can be found in appendix D and E. Associated with the NP tetrad are the spin coe cients, namely 12 complex scalars projections of the connection !IJ , e.g. (minus) the complex shear := the connection is Levi-Civita, these are functions of the metric. In the rst order formalism on the other hand, the connection is an independent variable, and the NP spin coe cients will be functions of the metric and of the connection components. To distinguish the two situations, we will keep the original NP notation, e.g. , for the Levi-Civita coe cients, and add an apex for the spin coe cients with an a ne o -shell connection, e.g. . On-shell of the torsion-less condition, !IJ = !IJ (e) and = . Explicit expressions for all the spin coe cients are in appendix A, and we will report in the main text only those relevant for m m r l .16 If the discussion. 4.2 The a ne null congruence Since the normal vector l is null, it would be automatically geodesic with respect to the spacetime Levi-Civita connection. Furthermore it would have vanishing non-a nity since 16The reader familiar with the NP formalism will notice an opposite sign in this de nition. This is a consequence of the fact that we work with mostly plus signature. and scalars, it is the unit normal to a null foliation. With an o -shell, a ne connection !IJ on the a other hand, these familiar properties do not hold. Using Newman-Penrose notation with an apex for the spin coe cients of the a ne o -shell connection, what we have is l r l = l m + cc; with `non-a nity' and `non-geodesicity' that are given respectively by 1 N k(l) := + cc = Ea i( ai i i = 1 N 2 i Ea i a : For the same reason, the congruence r l is not twist-free, even though l is the gradient of a scalar, nor de ned intrinsically on S: it also carries components away from it. Nonetheless, we can still take its projection ? ? r l , and decompose it into irreducible components: we will refer to the traceless-symmetric , trace and antisymmetric parts ! as `connection shear', `connection expansion', and `connection twist'. The components away from the hypersurface , which are all proportional to the shift vector N a, are not directly relevant for us and we leave them to appendix B. Using the de nition !IJ e J and the decomposition (3.14), we have for the hypersurface components HJEP1(207)5 In NP notation, shear, twist and expansion are described by the following two complex := := m m r l = m m r l = m m 1 2 (l) (l) = m m !(l) 1 N Ea = a ; 1 N E+a a ; where the real and imaginary parts of carry respectively the connection expansion and twist. It is also convenient to introduce the complex shear (l) := m m (l) = . This comes up awkwardly opposite in sign to the NP spin coe cient, but the minus sign is an unavoidable consequence of the fact that we work with mostly plus signature, the opposite to NP. The connection shear so computed allows us to identify Sachs' constraint-free initial data for rst-order general relativity in terms of real connection variables: in the absence of torsion, = and we can follow the same hierarchical integration scheme. From the connection perspective, the relevant piece of information is thus Ea a ; namely the contraction with the triad of a , which is the translation part of the ISO(2) stabilising the null direction xI+. Notice that both connection term and triad term have the same internal (4.8) (4.9) (4.10) (4.11) (4.12) (4.13) helicity: loosely speaking, it is this coherence that allows to reproduce the spin-2 behaviour in metric language. Notice that at the level of Poisson brackets, the shear components commute: trivially in f (l); (l)g = 0, but also when the conjugate appears, since17 (l); (l) = Im( ); 2i N E of (2.10). follows, which vanishes on-shell of the Gauss law, as we show in the next section. This is to be expected, since it is only at the level of the Dirac bracket that the shear components do not commute with themselves, that is when the light-cone constraints are used. We will show below in section 5.1 that the Dirac bracket reproduces exactly the metric structure In terms of the covariant connection, the shear, twist and expansion are described as (l) = eI eJ m m l !IJ ; = eI eJ m m l !JI : Using these covariant expressions, it is easy to see how the congruence is a ected by the presence of torsion, writing !IJ = !IJ (e) + CIJ where CIJ is the contorsion tensor. For instance, consider the case of fermions with a non-minimal coupling [58] S = i Z 4 e e I I (a ib 5)D (!) + cc; a; b 2 C; Re(a) 1: (4.16) (The minimal coupling would be a = 1, b = 0). Solving Cartan's equation, one gets (restoring for a moment Newton's constant G) CIJ = 2 eK G 2 where V I = I and AI = I 5 decomposition into (4.15) we nd Im(b)V L K [I Re(b)AJ] + Im(a)V J] (4.17) are the vectorial and axial currents. Plugging this h G in = ; = A Im(b)V l Re(b)A + Im(a)V (4.18) The connection shear recovers its usual metric expression, whereas twist is introduced proportional to the axial current; for non-minimal coupling, the twist depends also on the vectorial current, and furthermore the expansion is modi ed, picking up an extra term proportional to the time-like component of the vectorial and axial currents. More in general, for an arbitrary contorsion decomposed into its three irreducible components (3=2; 1=2) (1=2; 3=2) (1=2; 1=2) (1=2; 1=2), ; i : (4.14) (4.15) (4.19) (4.20) C ; = C ; + g [ C ] + 2 3 1 e = m m l C ; ; = m m l C ; + l C in C^ ; 17Using the brackets (3.19), and notice that f ai; Ejb=(N E)g = 1=(2N E)( ab ij EiaEbj=2). C^ ; 1 3 of [36], we expect the connection shear to be the conjugate momentum to the conformal metric. This expectation is indeed borne out, as we will show below in section 5.1. Summarising, the congruence generated by l is made geodesic by three rst-class Gauss constraints. The fourth rst-class one gives the relation between the connection expansion and the metric expansion. All these conditions are automatically preserved under evolution in u, since there are no secondary constraints arising from the stabilisation of the Gauss law. As for the connection shear, its relation to the metric shear is realised by the light-cone secondary simplicity constraints, and they are not automatically preserved. Tertiary constraints are required, to whose analysis we turn next. Tertiary constraints as the propagating equations Let us now discuss the tertiary constraints (3.23), whose presence is something quite unfamiliar within general relativity, and which is due to the combined use of a rst-order formalism and a null foliation: each feature taken individually introduces a secondary layer of constraints in the Hamiltonian structure. Perhaps even more surprising is which of the eld equations are described by these constraints: the propagating Einstein's equations, namely the dynamical equations describing the evolution (in retarded time u) of the shear away from the null hypersurface. In fact, it was shown in [24] that ab = 1 2N cadb h4g efh g0egcf (? GT)dh + Eic (Bdi + N dB0i)i ; where in the rst term we recognise the propagating Einstein's equations, and B I := e J F IJ (!) 0 denotes the algebraic Bianchi identities. This means that in the rst-order formalism, the only time derivative present in the propagating equations (3.10) can be completely encoded in algebraic Bianchi equations. The equivalence (4.29) may appear geometrically obscure, and it is furthermore not completely trivial to derive as a tensorial equation. On the other hand, it becomes transparent using the Newman-Penrose formalism, as we now show. To that end, let us rst identify the propagating equations in the Newman-Penrose formalism. A straightforward calculation of the propagating equations gives (4.29) (4.30) HJEP1(207)5 m m G (!; e) = m m G (!; e) = m m R (!; e) = Rlmnm(!; e) Rnmlm(!; e) 2Rlmmn(e); where in the last equality means on-shell of the torsion-less condition.22 Next, let us look at the tertiary constraints in its form (3.23), and project it in the same way on S: 1 2 mamb ab = mambEi(a b)ef Fe0fi j Feifj : (4.31) 22These equations are not be confused with Sachs' optical equations Rlmlm and Rlmlm, which relate Weyl and Ricci to the variation of shear and twist along the null hypersurface, not away from it. First, we have that Fe0fi(!; e) j Feifj (!; e) = n ei R ef (!; e): 2 N Then, to obtain the hypersurface Levi-Civita symbol, we observe that n is the only vector with a u-component, therefore we can write23 def = e6l[dmemf]: Finally, using the fact that maEiaei = m , we have 1 E condition, 1 e mamb ab = n m md def R ef (e; !) = 2n m l m R (e; !) = 2Rnmlm(e; !); which coincides with (minus) the propagating equations on-shell of the torsion-less It is also instructive to see the explicit role played by the algebraic Bianchi identity. For vanishing torsion and NP gauge,24 the propagating equation reads mamb ab Em m G (e): + + ( + 3 ) + 2 + 02 = 0; (4.32) (4.33) (4.34) (4.35) (4.36) HJEP1(207)5 where we can further set 02 = 0 since we are interested in the vacuum equations. Here := n r and := m r is conventional NP notation, see appendix A. For an expression of this equation in metric language, see e.g. [13]. The point is that if the connection is initially independent from the metric, this is a PDE with a single time derivative in the term ; but this term can be eliminated using an algebraic Bianchi identity, or `eliminant relation' in the terminology of [55]. Using equation (g) on page 48 of [55], which in NP gauge reads D 2 ( + ); (4.37) we can replace with ( + ) D plus squares of spin coe cients. In metric variables, this would indeed be a trivial manipulation, since the time derivative is now simply shifted from = + : : : to D = + : : :. But used in the rst order formalism with an independent connection (where now (4.37) holds with all quantities and it is derived from (4.30)), relates non-trivially the propagating equations to the tertiary constraint. Finally, concerning the geometric interpretation of this constraint, recall from section 3.2 that it is there to stabilise the light-cone conditions: hence, Einstein's propagating equations can be seen as the condition that a metric-compatible connection shear on the initial null slice, remains metric at later retarded times.25 23With conventions 0123 = 1, e = 1=4! IJKL eI eJ eKeL: 24Namely = , = = = 0, = + . See appendix D.2 for details. 25This can be compared with the metric formalism of [36], where the propagating Einstein's equations also arise from the stabilisation of the light-cone shear-metric conditions, but as multiplier equations, not as constraints. The discussion in the last two sections has been completely general: apart from the condition of having a null foliation, we have not speci ed further the coordinate system. We now specialise to Bondi coordinates, presenting the simpli ed formulas that one obtains in this case. We will then use this gauge to prove the equivalence of the symplectic potentials of the rst-order and metric formalisms, which in particular identi es the connection shear with the momentum conjugated to the conformal 2d metric; and to discuss a property of radiative data at I+. To that end, we completely x the internal gauge, adapting the doubly-null tetrad to a 2 + 2 foliation. For the interested reader, the Bondi gauge for our tetrad without the complete internal gauge- xing is described in appendix E.1. We take i = (1; 0; 0) as in (4.5), and use the rst-class generators K and TM to x Eir = (1; 0; 0). This internal `radial gauge' adapts the tetrad to the 2 + 1 foliation of constant-r slices: ) Ea1 = (1; 0; 0); E = p ; m = (0; 0; EA): (5.1) The determinant of the triad now coincides with that of the induced metric AB (hence triad and metric densities now conveniently coincide). This xes ve of the internal transformations, leaving us with the SO(2) freedom of rotations in the 2d plane of mappings m 7! ei m . We will not use this freedom in the following, and if desired can be xed for instance requiring the triad to be lower-triangular. Now we impose the coordinate gauge- xing. On top of the null foliation condition g00 = 0, the Bondi gauge conditions are g0A = 0, plus a condition on r, typically either the areal choice p = r2f ( ; ), or the a ne choice g01 = 1. We take here the a ne Bondi gauge, and report the details on Sachs areal gauge in appendix E.2. From the parametrisation (3.6), we can read these conditions in terms of our tetrad variables: g0a = Ea i = ( 1; 0; 0): i Using the internal gauge- xing (5.1), Er i = E1r = 1, hence (5.2) implies E1A = 0 and i N = 1, as in the metric formalism. The metric (3.5) and its inverse reduce to the following form, Eai = 1 0 EAM E1A EAM Eia = 1 0 where as before we use M = 2; 3 for the internal hypersurface coordinates orthogonal to i, and EMA is the inverse of the dyad EAM . 0 2U + ABN AN B ABN B1 g = B C ; A g 0 0 = B 1 2U N A C ; 1 0 ABA where we rede ned 2U := 1 2N r for convenience. The triad and its inverse are (5.2) (5.3) (5.4) 1 0 1 N 0 AB ! ; The structure of the null congruence of l reduces to: k(l) = 1 r ; (l)AB = (MAEB)M ; The lapse equation (4.26) simpli es to !(l)AB = [MA EB]M : = r ; (l) = AM EMA ; r1 = 0; [AB] := [MA EBM] = 0: so this connection component is set to zero by working with a constant lapse. The vanishing of the twist imposed by L (in absence of torsion) now reads HJEP1(207)5 This equation is the null-hypersurface analogue of the familiar symmetry of the extrinsic curvature in the spatial hypersurface case, there analogously imposed by part of the Gauss constraint: K[ab] := K[iaEbi] = 0. The radial boost K simpli es to (5.5) (5.6) (5.7) (5.8) (5.9) (5.10) K = p ; and its solutions give the a ne Bondi-gauge formula for the expansion, (l) = @r ln p . The solution of the light-cone secondary simplicity constraints (4.27) now gives namely the expression for the shear in a ne Bondi gauge, written here in terms of the dyad EMA . 5.1 Equivalence of symplectic potentials We now show the equivalence between the symplectic potential in connection variables (which we can read from the p q part of (3.15)) and the one in metric variables (2.9), thereby identifying the canonical momentum to the conformal 2d metric in the connection language. It will turn out to be the connection shear of the canonical normal n1 = N l , as to be expected from the on-shell equivalence of the rst and second order pure gravity action principles. As in the usual space-like canonical analysis, the equivalence of symplectic potentials will require the Gauss law. We begin by eliminating i and ij from the phase space, completely xing the internal gauge and using the primary simplicity constraints, and consider then only the rst term of (3.15) for the symplectic potential. Since our main focus are the bulk physical data, we will neglect boundary contributions to the symplectic potential, and show the equivalence in the partial Bondi gauge g0A = 0. The reason not to x completely the Bondi gauge is to keep both lapse and an arbitrary p , to show a more general equivalence holding regardless of the choice of coordinate r. Hence, we want to show that with AB given by (2.6). = Z 2Eia i (g0A=0) Z a p AB AB; (5.11) The partial Bondi gauge is EA i = E1A = 0, which implies r i Gauss law, see (5.5). This eliminates two monomials from the integrand, and we are left with the following two terms: = Z Accordingly, here and in the following we will restrict attention to variations preserving the gauge and the Gauss constraint surface. Let us look at the right-hand side of (5.11). We expect from the metric formalism that the conjugate momentum is build from the congruence of n1 = N l . Its shear and expansion are just N times those of l , which we can read from (4.11); its non-a nity is k(n1) = @r ln N = r1 using the lapse equation (4.26) in partial Bondi gauge. Accordingly, we consider the following ansatz for the momentum, (5.12) (5.13) (5.14) (5.15) (5.16) (5.17) AB := AM EBM AB(EMA AM + r1); whose decomposition gives AB 1 (n1) 2 r1; where we used r M = 0 = [MA EB]M from the Gauss law. This momentum reduces to the one in the metric formalism (2.6) by construction, and we now show it satis es (5.11). To that end, we rst observe that E = p is now a 2 2 determinant. This means that det EAM = det EMA , and the inverse induced metric has the following expression in terms of canonical variables, AB = EMA EBM (det EMA )2 : A simple calculation then gives AB AB = = AB AB = 2 (AB)EBM EAM EMA 2 h (MAEB)M EBN ENA + r1EAM EMA i : where we used det E = EAM EMA . Next, we use again [MA EB]M = 0 from the Gauss law, so the rst symmetrised term above gives twice the same contribution. Using the fact that p = N EAM EMA , we nally get AB AB 2 M EMA + r1 p A ; and (5.11) follows up to boundary terms. We have thus veri ed that in the rst order formalism the (traceless part of the) conjugate momentum to the induced metric is the connection shear of n1 = N l . We also remark the presence of a term proportional to the 2d area. As in the metric formalism, this is a measure-zero degree of freedom, that can be pushed to a corner contribution and describes one of Sachs' corner data. A similar corner term appears in the spinorial construction of [23], where it is shown to admit a quantisation compatible with that of the loop quantum gravity area operator. See also [59] for related results on This result provides an answer to one of the open questions of [24], namely that of identifying the Dirac brackets for the reduced phase space variables. We did so looking at the symplectic potential as in covariant phase space methods, and completely xing the gauge: this introduced additional second class constraints that could be easily solved, e.g (5.7). Whether it is possible to write covariant Dirac brackets without a complete gaugexing remains an open and di cult question, because of the non-trivial eld equations satis ed by the second class Lagrange multipliers. It is interesting to compare the situation with the space-like case, where the dynamical part of the connection is also contained in components of ai, except now i belongs to a time-like 4-vector (and we can always set i = 0, a choice often referred to as `time gauge', since e0 / dt). These dynamical components describe boosts and therefore do not form a group. An SU(2) group structure can be obtained via a canonical transformation, to either complex self-dual variables, as in the original formulation [ 19 ], or to the auxiliary Ashtekar-Barbero real SU(2) connection (see e.g. [60]): the transformation requires adding the Immirzi term to the action, and the price to pay is either additional reality conditions, or use of an auxiliary object instead of a proper spacetime connection. Using a null foliation appears to improve the situation: the three internal components of ai can be naively26 associated with the radial boost K and the two `translations' T i , or null rotations, related to the ISO(2) group stabilising the null direction of the hypersurface. But as we have seen ? above only the translation components aM enter the bulk physical degrees of freedom, which are described by the connection shear. The component ai i is on a di erent footing: it enters the spin coe cients ; ; and (see appendix A), and is treated in a way similar to the expansion , in that it is fully determined from initial data on a corner. We plan to develop these ideas in future research, in particular investigating the relation with a loop quantum gravity quantization based on the translation components of the connection, representing bulk physical degrees of freedom. To complete the comparison between null and space-like foliations, in the latter case the canonical momentum conjugated to the induced metric is build from the triad projection Kai of the extrinsic curvature (see e.g. [60] for details). For a null foliation, the canonical momentum conjugated to the induced metric is related to the shear of the null congruence. The comparison is summarised by the following table:27 Foliation Relevant internal group Momentum conjugated to metric Space-like SU(2) Null ISO(2) ab = KaiEbi qabKciEic ab = Xij( aiEbj qab ciEcj) To help the comparison in the table above, we have used the fact that in our formalism we can de ne the raised-indices hypersurface metric qab, and use it to prescribe an extension ab of (5.13) on the whole hypersurface.28 26To make the argument precise, we should embed the dynamical components into a covariant connection whose non-dynamical parts are put to zero by linear combinations of constraints, see e.g. [61] for an analogue treatment in the space-like case. 27For the reader interested in the time-like case, see [62]. 28The equivalence (5.11) can then be written with ab qab on the right-hand side, and trivially holds because the extra pieces now present are put to zero by the constraints and/or gauge conditions. HJEP1(207)5 face of the foliation attached to future null in nity, plus the asymptotic transverse shear s0(u; ; ). Thanks to the algebraic Bianchi identity (4.37), this can also be understood as prescribing a certain shear for the non-geodetic asymptotic null vector @u @r in the physical spacetime. Radiative data at future null in nity as shear `aligned' to I As a nal consideration, we would like to come back to the geometric interpretation of the tertiary constraints, and point out that the very same algebraic Bianchi identity that links them to the propagating equations, also plays an interesting role in the interpretation of the radiative data at I+. To that end, we consider in this subsection the case of an asymptotically at spacetime, and the u =constant null foliation attached to future null in nity I+. In this setting, we can compare our metric (5.3) and doubly-null tetrad to those of Newman-Unti [56, 57, 64] mostly used in the literature, and use the asymptotic fall-o conditions for the spin coe cients there computed.29 We refer the interested reader to appendix E.1 for the details, and report here only the most relevant results. In particular, 0 = r2 + O(r 4); (5.18) and the asymptotic shear 0(u; ; ) fully characterises the radiative data at I+ [5, 7, 10]. Ashtekar's result [10] (see also [12] for a recent review) is that the data can be described in terms of a connection D D l 1 2 de ned intrinsically on I+, related to the shear by 0 = D l . This description has led to a deeper understanding of the physics of I in this paper. future null in nity, showing among other things that the phase space at I (there is no super-translational invariant classical vacuum). The connection description at + inspired and is exactly analogous to the local spacetime connection description studied + is an a ne space From the perspective of the 2+2 characteristic initial-value formulation (with backward of I evolution | or we should rather say nal-value formulation), this means that one can think + as one of the two null hypersurfaces, but the relevant datum there is not the shear along it (which vanishes!), but the transverse asymptotic shear 0(u; ; ) at varying u, see gure 2. However, we now show that thanks to the Bianchi identity (4.37), this datum can also be identi ed as shear of a vector eld in the physical spacetime. 29In using these results, care should be taken in that the authors use a slightly di erent de nition of coordinates: u is now 1=p2 the retarded time, and r is p 2 the radius of the asymptotically at 2-sphere. but non-geodesic, with In the asymptotic expansion, n r n = n + m + cc: is leading-order twist-free and a ne, but still non-geodesic: !(n) := Im( ) = m m r n = O(r 2); = O(r 2); = 3 + O(r 2): (5.21) The non-geodesicity at leading order depends on one of the asymptotic complex projections of the Weyl tensor, in turn given by the radiative data 30 = _ 0.30 Since n is not geodesic, it is also not hypersurface orthogonal, in spite of being twist-free at lowest order: the radiative term _ 0 prevents the identi cation of a null hypersurface normal to (5.20) (except in the very special case of completely isotropic radiation at all times). Consequently, there is no unique de nition of shear for the congruence it generates. Using the NP formalism, it is natural to consider the shear along the 2d space-like hypersurface spanned by m , 0 To that end, consider the second null vector of the tetrad, n . It is null everywhere (5.19) (5.20) (5.22) (5.23) and de ne to give (n) := = m m r n = 0 + O(r 2): 0 = _ 0; At the same lowest order O(r 1), the algebraic Bianchi identity (4.37) can be solved which relates the transverse asymptotic shear to the -shear of n . Hence, the radiative data at future null in nity correspond to a shear of a non-geodesic vector eld `aligned' with I+. The fact that the vector is non-geodesic shows that the asymptotic 2 + 2 problem can not be formulated in real spacetime. On the other hand, this is how close one can get, in terms of the interpretation of the main constraint-free data, in bridging between the local 2+2 characteristic initial-value problem, and the asymptotic one. 6 Conclusions In this paper we have presented and discussed many aspects of the canonical structure of general relativity in real connection variables on null hypersurfaces. We have clari ed the geometric structure of the Hamiltonian analysis presented in [24], explaining the role of the various constraints and their geometric e ect on a null congruence. We have seen how the Lorentz transformations of the null tetrad are generated canonically, and how to restrict them so to adapt the tetrad to a 2 + 2 foliation, and compare the connection Hamiltonian analysis to the metric one. Lack of canonical normalisation for a null vector means that the equivalence of the lapse functions can only be given up to a boost along the null direction. Restricting to the Bondi gauge, we have identi ed constraint-free data in connection variables, and shown equivalence of the symplectic potential with the metric 30This can be seen solving at rst order in 1=r the NP components Rnmnl and Rnmnm of the Riemann tensor, see e.g. (310i) and (310m) of [55]. formalism. The metric canonical conjugated pair `conformal 2d metric/shear' is replaced in the rst order formalism by a pair `densitized dyad/null rotation components of the connection', with the null rotations becoming the shear on-shell of the light-cone secondary simplicity constraints. In the presence of torsion, the connection can pick up additional terms that contribute to the shear, twist and expansion of the congruence, leading to modi cations of Sachs' optical and Raychaudhuri's equations. Even in the absence of torsion, the on-shell-ness is not automatically preserved under retarded time evolution, but requires of tertiary constraints, something unusual in canonical formulations of general relativity. We have shown that the tertiary constraints encode Sachs' propagating equations thanks to a speci c algebraic Bianchi identity, the same one that allows one to switch the interpretation of the radiative data at I transverse asymptotic shear 0 to the `shear' 0 of a non-geodetic, yet twist-free, null + from the vector aligned with I+, suggesting a di erent perspective on the asymptotic evolution problem. The identi cation of the connection constraint-free data as null rotations means that the degrees of freedom form a group, albeit non-compact, hence one could try to use loop quantum gravity quantization techniques without introducing the Immirzi parameter. Some of the corner data, which we did not investigate here, have already be shown to lead to a quantization of the area [22, 50, 59]. A quantization of the connection description of the radiative degrees of freedom can lead to new insights both for loop quantum gravity and for asymptotic quantisations based on a Fock space. We completed the paper with an extensive appendix, presenting the explicit calculations of the rst-order spin coe cients for the tetrad description used, and a detailed comparison between null tetrad descriptions and 2 + 2 foliations. We hope that the connection formalism can provide a new angle on some of the open questions on the dynamics of null hypersurfaces in general relativity, and we plan to come back in future research to some of the important aspects left open here: in particular, investigating the symplectic potential and Dirac brackets among physical data without the Bondi gauge, as well as including boundary terms and identifying the BMS generators in this Hamiltonian language. We also plan to develop further the indications that the connection degrees of freedom now form a group and its possible applications to quantisation. Acknowledgments We are indebted to Sergei Alexandrov for many exchanges and a careful reading of the draft. We also would like to thank Abhay Ashtekar, Tommaso De Lorenzo, Michael Reisenberger and Wolfgang Wieland for helpful discussions. A Spin coe cients We use i = (1; 0; 0) and v := (v2 iv3)=p2. For the tetrad derivatives we have D = l r 1 N E1ara; = m r = Ea ra; = n r = m r = 1 2 rt = E+ara : N a + E1a ra ; (A.1) (A.2) 1 2 1 2 1 2 1 2 (n l + m m ) = (n l + m m ) = l + m m ) = (n Dl + m Dm ) = m Dl = m l = m l = m l = 1 N 2 E1a a 21 E1a a + N N 1 E+a a a 12 E+a a1 1 Ea 1 a 2 1 4 4 1 2N a N E1a a1 E1a a1 p N 2 (!00 N a N N Ea + a 1 ir22 ir33 i 2 r1+ 2 r1 + 4 ! 1 !+ 4 1 2 1 N a a1 + 2N r11 2 1 2 1 2 1 2 ln N ln N N r11 + D ln N !01 ) := := := := := := := := (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) (A.9) (A.10) (A.11) (A.12) (A.13) (A.14) (A.16) (A.17) := m n = := m n = := m n = := m Dn = 1 2 1 2 N 4 N 2 1 2A A; 7! 1 1 k 7! A2 k; 7! A 7! A2 ; 7! 7! ; 7! A ; 7! A ; (N E1a + 2N a) a+ N a(Ea1!+ Ea+!1 2iEairi+) + p N (!00+ + !01+) 1 2 2 N E+a a+ + 2r23 ir22 + ir33 21 E1a a+ + 1 !+ 2 ir1 For the spin coe cients we use the standard notation consistent with our mostly plus signature (which carries an opposite sign as to the notation with mostly minus signature) and use an apex to keep track of the fact that the connection !IJ is o -shell. We then have i N a(Eal 1lm!m) + 1 (!001 i!023) 2 i N aEajr1j 1 2 ln N Under the rescaling (l ; n ) 7! (l =A; An ) (a class III transformation), 2A A; 1 2A A; 1 7! A 1 2A DA; (A.15) 7! A 7! A 1 7! ; ; Hence, many factors of N disappear in the spin coe cients if we use the ADM-like normal Congruence The complete expression of the congruence tensor with an a ne connection is 1 !ij N (!00j Xij Eai + !ij j Eai); 0 1 N 1 0 j EaiN a + ral0 = ai N Xij N aEaj N Xij aiEbj ; N N B0a = N 1 qbcN c bM EaM ; Ba0 = N 1 qbaN c bM EcM ; r l given by B00 := Bab = 1 qcbN aN b aM EcM ; N 1 qcaqdbXij aiEbj : C Tetrad transformations and gauge xings At the Hamiltonian level, the Lorentz transformations are generated by the Gauss constraint GIJ , usually decomposed into spatial rotations Li and boosts Ki, whose canonical form from (3.16) reads Li := 1 2 ijkGjk = @a( ijkEja k Eja i) aj Xij !~j : ij k ajEka ij k!~j k; Since we are working on a null hypersurface, it is convenient to introduce the subgroups ISO(2) stabilising the null directions x I = ( 1; i), with generators T I := 1=2 IJKLx+J JKL and T^I := 1=2 IJKLx J JKL. Both groups are 3-dimensional and con? tain the helicity generator L , plus two independent pairs of `translations', T i := ijk j Tk stabilising xI+, and T^i := ijk j T^k stabilising xI . Taking both sets and the radial boost ? K we obtain the complete the Lorentz algebra, expressed in terms of canonical variables in (4.22). For ease of notation and to make the formulas more transparent, we x from now on i = (1; 0; 0), as we did in most of the main text. We use the orthogonal internal indices M = 2; 3, and write the canonical form of the generators as follows, L1 = 1MN EaM N a ; EMa aM ; TM = ^ TM = 1MiE1a ai 1Mi(E1a ai To compute the action on the tetrad, we use the brackets (3.19). First of all, T^M change the internal null direction i : Since the direction is gauge- xed by (3.18) in the action, these constraints are second class. The stabilisers TM are rst class, and can be used to put the triad in (partially) lower (B.2) (C.1) (C.2) (C.3) (C.4) invertibility of the triad. The radial boost K can be used to x E1r = 1, since so we can always reach EMr = 0 with these transformations, and EA1 = 0 follows from the fK1; E1rg = 0; fK ; Eg = 12 E; fK ; E1rg = 12 E1r: The triad so gauge- xed reads Eai = 1 0 ! Eia = 1 E1A EMA 0 ! ; (C.5) (C.6) (C.7) (C.8) (C.9) triangular gauge of the triad. triangular form: p g = N E = N p . Finally, the helicity rotation L1, acting as so the coordinates are adapted to the 2 + 2 foliation. Furthermore, E = p where EAM is the 2d dyad with inverse EMA , and E1A = EMA ErM . In this gauge, d 1 = dr, and so fL1; Eiag = 12 1MiEia; can be used to put to zero one o -diagonal component of the dyad and thus complete the Using hypersurface di eomorphisms instead, we can put the triad in (partially) upperfD(N~ ); Eiag = $N~ Eia; so we can use DA to x EA = 0, and Dr to x E1r = 1. This gives Eai = 1 EA1 ! Eia = 1 EMr ! with EMr = EMA EA1. In this gauge the hypersurface coordinates are not adapted to the 2 + 1 foliation (the level sets r =constant do not span the 2d space-like surfaces), on the other hand the tangent to the null directions is now the coordinate vector @r. For clarity, the various conditions that can be xed using the various constraints are summarised in the table below, where by rgf we mean the nal gauge xing on r, for instance a ne or areal. H g00 = 0 T^i ? i = (1; 0; 0) DA T i ? EA = 0 , ErM = 0 Er = 1 aut rgf EMr = 0 , EA = 0 rgf aut Er = 1 L Dr K Notice that if one does not x the upper or lower triangular form of the triad, the inverse of the 2d dyad if of course not given by the corresponding entries of the inverse triad. A general parametrisation of the triad in terms of the dyad can be easily written as follows, Eai = M EAM fM M EA 1 fM ! (C.10) Eia = 1 M The Bondi gauge sets A = 0, namely qra = 0. D 2 + 2 foliations and NP tetrads We collect here various useful formulas relating the tetrad formalism to the 2 + 2 foliation of [35] and [36]. As brie y explained in section 2.2, the 2 + 2 foliation is induced by two closed 1-forms, n := d locally, = 0; 1. These de ne a `lapse matrix' N , as the inverse of N := n n , and a dual basis of vectors n := N n1 are tangent respectively to the hypersurfaces 1 = const and 0 = const: We assume g n . Note that n0 and det N is ? < 0, so that the codimension-2 leaves fSg are space-like. The projector on fSg N n , and the covariant induced metric :=? . The 2d spaces fT g tangent to n are not integrable in generic spacetimes, since ? [n0; n1] 6= 0. This non-integrability is often referred to as twist in the literature. On the other hand, the orthogonal 2d spaces foliate spacetime by construction, and we can introduce shift vectors to relate the tangent vectors to coordinate vectors, b To write the metric explicitly, we take coordinates ( ; A) adapted to the foliaN + ABbAbB AC bC BC bC ! AB g N N N b B b A AB + N M function. Then AB = EA EMB and Here EAM is the dyad and EMA its inverse, E = E M and M = M^ qab = AB A B BA A AB B ! AB : (C.11) : (D.1) (D.2) (D.3) (D.4) (D.5) 1 2 (n0; n~1) with For a null foliation, we x one di eomorphism requiring N11 = 0 = N 00 = g00, so that the rst normal is null, and N 01 = 1=N01, N 11 = N00=N021. The norm of n1 is N 11 and we leave it free (it can be both time-like or space-like without changing the fact that the orthogonal spaces fSg are space-like), but notice that we can always switch to a null frame n~1 = N01n1 + N00n0; jjn~1jj2 = 0; n0 n~1 = 1: This can be used to de ne the rst two vectors of a NP tetrad adapted to the foliation, via n0 , n := n~1 , so that the 2d space-like induced metrics coincide = g N n n = g + 2l( n ): Notice that acting with a Lorentz transformation preserving l, we have n 7! n + am + am + jaj2l ; m 7! m + al ; one thus obtains a new covariant 2d metric, still space-like and transverse to l , but not associated with the 2+2 foliation any longer. In terms of the NP tetrad, the non-integrability of the time-like spaces is measured by the two spin coe cients and , m [l; n] = + : We can also reverse the procedure: start from an arbitrary NP tetrad, and adapt it to a 2 + 2 foliation. To that end, recall rst that )l + ( )n + ( )m ; (D.6) so the general non-integrability of (m; m) is given by non-vanishing Im( ) and Im( ). To adapt the NP to the 3 + 1 null foliation, we choose l := d 0. This xes 3 Lorentz transformation, and implies Im( ) = 0 = and = + . We can also x the SO(2) helicity rotation requiring = . This leaves us with two tetrad transformations left. To HJEP1(207)5 have a 2 + 2 foliation induced by the tetrad, we need This is achieved if in coordinates ( ; A) adapted to the foliation m = (0; 0; mA), hence = (c ; 0; 0) by orthogonality; this xes the remaining two tetrad freedoms (And if we x radial di eomorphisms to have N 01 = 1, this gauge also implies = + ). Inverting this linear system we nd d 0 = c0 l + c1 1 This identi es c = (N00=2; N01), and (D.3) follows again. For more on the characteristic initial value problem in NP formalism see e.g. [65]. The use of a tetrad adapted to a 2 + 2 foliation is common, e.g. [13, 66], but not universal. In particular in [1] the partial Bondi gauge is completed with N 11 = 0 = N00 = c0, so to have both 1-forms d null. D.2 The Bondi gauge and Newman-Unti tetrad A more wide-spread tetrad description, particularly suited to study asymptotic radiation, is the one introduced by Newman and Unti [64], see e.g. [56, 57] for reviews, which is adapted to the 3 + 1 null foliation and to the Bondi gauge. We take coordinates (u; r; ; ) and x g00 = 0, so that the level sets of u give a null foliation with normal l = the null hypersurfaces normal to l are ruled by null geodesics, with tangent vector N This suggests a natural 2 + 1 foliation of given by the level sets of a parameter along the null geodesics (a ne or not). The description simpli es greatly if we gauge- x g0A = 0, as to identify the geodesic parameter with the coordinate r, while simultaneously putting to zero the shift vector of the r = const: foliation on . In other words, the (partial) Bondi gauge g0A = 0 gives a physical meaning to the coordinate foliation de ned by u and r by identifying it with the foliation de ned by the null geodesics on . In the 2 + 2 language of [35, 36], with adapted coordinate 0 = u, the gauge corresponds to a vanishing shift { 35 { (D.7) (D.8) (D.9) Let us complete the Bondi gauge choosing a ne parametrization, namely g01 = The metric and its inverse read g 0 0 1 0 1 g11 g1A C ; g11 + gABg1Ag1B 1 gABg1B1 0 C : A The Newman-Unti tetrad adapted to these coordinates is chosen identifying l with the normal to the foliation, and requiring n and m to be parallel propagated along l . It is parametrised as follows, (D.11) (D.12) with A = ; stereographic coordinates for S2 ( = cot =2ei ), and g11 = 2(j!j2 U ); g1A = ! A + ! A XA; gAB = A B + A B : The co-tetrad is l = ( 1; 0; 0; 0); n = U = ( gAB AXB; 0; gAB B): gABXA(! B + ! B); 1; gAB(! B + ! B ) ; A 7! e The coe cients are a priori 9 real functions (U 2 R; XA 2 R2; ! 2 C; A 2 C2) parametrising the 6 independent components of the metric plus 3 internal components corresponding to the ISO(2) stabiliser of l . The helicity subgroup generates dyad rotations i A, and the translations the class I transformations (D.4). The latter in particular shift ! 7! ! + a, a 2 C, and can be used to put ! = 0, so m components only. This is the 2 + 2-adapted choice described above, and corresponds to EMr = 0 as in the lower-triangular form (C.6), that we also used in section 5 in the main text to make easier contact with the metric Hamiltonian formalism. Alternatively, this null rotation can be used to achieve = 0, so to make n and m to be parallel propagated = (0; 0; mA) with 2d space-like along l as demanded by Newman and Unti. In terms of spin coe cients, we have the following simpli cations: = Im( ) = 0; which follow from l being a gradient, Re( ) = 0 from xing the radial di eos requiring r a ne parametrization, and = 0 from the parallel transport of n and m . Finally Im( ) = 0 if we x the helicity SO(2) rotation. This complete xing is usually referred to as NP gauge, to be contrasted with the 2 + 2-adapted gauge described above, where the condition = 0 is replaced by and Im( ) = 0. Hence, when we refer to the Newman-Unti tetrad (D.11) in NP gauge there are only 6 free functions of all 4 coordinates. The NP gauge is preserved by class I and helicity transformations with r-independent parameters. E Mappings to the -tetrad In this appendix we discuss the detailed relation between the -tetrad used to perform the canonical analysis in real connection variables and the results of the previous appendix. It provides formulas completing the discussion in the main text. At the end of section (3.1) we introduced the internal `radial gauge' (3.13), stating that it adapts the tetrad to the 2 + 2 foliation and identi es the lapse function with the one used in the metric formalism. We now provide the relevant details and proofs. The -tetrad and its inverse are given by eI = N N aEai Eai i Eai eI = 1 N 1 N a N Eia + N a i i ! ; where ^ 2 = 1 to have a null foliation, e = EN and N = N function. Taking the soldered internal null directions x N aEai i is the lapse = eI x I of (3.7), and de ning ie3 ) when i = (1; 0; 0), the basis m to be a complex linear combination of the two orthogonal tetrad directions X ij ej , e.g. (E.2) (E.3) (E.4) (E.5) (E.6) is a doubly-null tetrad. We then rescale it by (x+; x ; m ; m ) l = 1 N n = x ; N 2 to de ne an NP tetrad adapted to the 3 + 1 null foliation as described in the main text, see (4.6). In general, the 2d spaces with tangent vectors (m ; m ) will not be integrable. With reference to (D.6), we see that integrability requires Im( ) = Im( ) = 0. The rst condition is guaranteed by the fact that l is a gradient. The second can be obtained with a class I transformation, generated by the translations Xij T j stabilising l , xing In this gauge EA = 0 , X ij Ejr = 0: = (0; 0; EA); n = (N (N=2 + N rEr ); N Er ; 0; 0); so that the null tetrad is manifestly adapted to the 2+2 foliation de ned by the level sets of the coordinates u and r. Then Im( ) = 0 also follows immediately by explicit calculation of (D.7) using the fact that m only has 2d surface components. In a rst-order formalism with independent connection, the statement holds in the absence of torsion. We have already seen in section 4.2 that on-shell of the torsionless condition Im( ) = Im( ). Let us show here explicitly how Im( ) goes on-shell. From (A.11) we have and from one of the secondary simplicity constraints (3.20) we have The last term vanishes for EMr = 0 = EA1, hence Im( ) = 0 in this gauge. N 2 2 N 2 Im( ) = Im( ) (r22 + r33); 11 = r22 r33 1MN EMa Eb1@aENb : To complete the comparison with the 2 + 2 formalism, let us x the internal direction i = (1; 0; 0), and use M = 2; 3 to refer to the orthogonal directions. Then (E.3) puts the triad in the form Eai = Er1 0 Eia = E1r 0 ! E1A EMA ; thus EAM is the 2d dyad and EMA its inverse, and we further have the equalities Er1 = 1=E1r, ErM = EAM E1A=E1r. We then have gAB = qAB = EAM EBM = AB = EA EBM , consistently M with the fact that the metric induced by the dyad is adapted to the coordinates by the gauge- xing, and qAB = EMA EBM = gauge g0A = 0 achieves gAB = AB is its inverse. Notice that the (partial) Bondi AB, analogously to the vanishing-shift gauge for spaceAt this point E = Er p and p g = Er N p . A look at the metric shows that 1=g01 = Er N; hence, the lapse function in the metric Hamiltonian analysis of [36] equals the one in the connection formulation up to a factor Er . This ambiguity is not surprising due to the null nature of the foliation and the lack of a canonical normalization of its normal. To identify our lapse with the one in the metric formalism is su cient to x the radial boost K as to have Er takes the form (C.6). We also recover the relation p g = N p between lapse and the = 1, as we did with (3.13). Then also Er = 1 because of (E.3) and the triad determinant of the metric Hamiltonian analysis. For completeness, we report below the relation between the -tetrad coe cients and the 2+2 foliation with a general radial gauge. The case with coinciding lapse functions can immediately be read plugging Er = 1 = Er in the formulas below. The relation between the foliating normals and the adapted null co-frame is given by n0 = du = x+I eI ; n1 = dr = 1 2Er N N +2N rEr x+I +x I eI ; n0 n1 = l + n = (1; 0; N rEr EA N A); b0A = N A N rEr EA ; (E.7) (E.8) 1 Er N : (E.9) (E.10) (E.12) (E.13) b1A = Er EA ; (E.11) 1 N 0 The dual basis, shift vectors and lapse matrix are n0 = N N r Er N 2 n1 = Er N l = (0; 1; Er EA ); N N N (N + 2N rEr ) N Er N Er 0 1 1 NEr N(Er )2 (N + 2N rEr ) 1 NEr ! ; ! ; and the formulas for the 2d projector and covariant induced metric coincide, := g x+( x ) = g N n n = qabN aN b qbcN c ! of the second shift vector in terms of ErM . EMA ErM , which provides an alternative characterisation The non-integrability of the fT g surfaces is the same as measured by the null dyad, [n0; n1] [n0; n1] ; m [n0; n1] = N ( + ): (E.14) Having gauge- xed N 00 = 0 to have du null and r a ne or areal, we cannot for general metrics simultaneously take dr to be null. It can be made null on a single hypersurface ~ de ned by some xed value of r = r0, if we exploit the left-over freedom of hypersurface di eomorphisms to x N 11 = 0. This is what was done by Sachs in setting up the 2 + 2 characteristic initial value problem, further xing N A = 0 on the same hypersurface, so that the normal vector of ~ at r = r0 is just n0 = n = @u, as in gure 1. E.1 The Bondi gauge and Newman-Unti tetrad In section 5 in the main text we discussed the Bondi gauge with a null tetrad already adapted to the 2 + 2 foliation. This was motivated by the goal of recovering properties of the metric symplectic formalism. On the other hand, the Newman-Unti tetrad (D.11) mostly used in the literature is adapted to the 3 + 1 null foliation only. In this appendix we present the relation between our metric coe cients and those of (D.11) without xing the internal `radial gauge' (5.1). To that end, we rst x all di eomorphisms requiring the Bondi gauge 1 N Ea = (1; 0; 0): (E.15) We then x the internal direction i = (1; 0; 0), and adapt l = du = x+=N . This leaves the freedom of acting with the ISO(2) subgroup stabilizing the direction. Because we rescaled the canonical tetrad by N , we also gain the freedom of canonical transformations corresponding to the radial boost K , which does not a ect l. This additional gauge freedom should be xed requiring Er = 1, implying N = 1. We are then left with 9 free functions, 6 for the metric and 3 for the internal ISO(2) stabilising l. Comparing our tetrad (4.6) in this gauge with (D.11) we immediately identify 1 2 U = N r; XA = N A; ! = Er ; A = EA: (E.16) g 0 = B The 2 + 2-adapted tetrad is recovered with a class I transformation setting ! = EM = 0. E.2 Areal r and Sachs' metric coe cients Above we used a ne r, as usual in literature using the Newman-Penrose formalism. The alternative common choice is Sachs', leaving g01 = r2f ( ; ). Again we x the internal direction i = (1; 0; 0) and the radial boosts with Er = 1, so to have the identi cation of our N > 0 with the metric lapse e2 . The triad e 2 free and requiring instead p = has the form (C.9), and the metric reads N (N + 2N r + 2N AEA ) + ABN AN B ABN B C : A (E.17) Comparing with (2.1) in the main text, we 1 2 V r ln N; U A = N A + N AB EB ; = 2N 1 + N (1 + AB EA EB ): Reverting to a ne r, N = 1 and the map from Sachs' metric coe cients to Newman-Unti's is V =r = 2(j!j2 U ), U A = X A ! A ! A : Open Access. Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [1] R.K. Sachs, On the characteristic initial value problem in gravitational theory, J. Math. Phys. 3 (1962) 908 [INSPIRE]. 21 [INSPIRE]. coe cients, J. Math. Phys. 3 (1962) 566 [INSPIRE]. relativity, Gen. Rel. 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Elena De Paoli, Simone Speziale. Sachs’ free data in real connection variables, Journal of High Energy Physics, 2017, 205, DOI: 10.1007/JHEP11(2017)205