#### 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.
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