Ehlers, Carroll, Charges and Dual Charges

We unravel the boundary manifestation of Ehlers' hidden M\"obius symmetry present in four-dimensional Ricci-flat spacetimes that enjoy a time-like isometry and are Petrov-algebraic. This is achieved in a designated gauge, shaped in the spirit of flat holography, where the Carrollian three-dimensional nature of the null conformal boundary is manifest and covariantly implemented. The action of the M\"obius group is local on the space of Carrollian boundary data, among which the Carrollian Cotton tensor plays a predominent role. The Carrollian and Weyl geometric tools introduced for shaping an appropriate gauge, as well as the boundary conformal group, which is $\text{BMS}_4$, allow to define electric/magnetic, leading/subleading towers of charges directly from the boundary Carrollian dynamics and explore their behaviour under the action of the M\"obius duality group.


Introduction
Hidden symmetries have a long history in relativistic theories of gravity, which started with the seminal work of Ehlers in the late fifties [ ]. It was shown in this article that in the presence of an isometry, vacuum Einstein's equations were invariant under Möbius transformations. This observation triggered an important activity in several directions. In line with the sixties' renaissance of general relativity, it opened the way for solution-generating techniques applicable to vacuum Einstein's equations [ , ].
This was soon generalized to situations with more commuting Killing fields [ , ] and bigger hidden symmetry group, providing the system with remarkable and unexpected integrability properties [ -].
The underlying deep origin for the above pattern was unravelled with the advent of higher-dimensional supergravity theories, and is rooted in the reduction mechanism. This has revealed a wide class of hidden groups, among which the exceptional play a prominent role (see e.g. [ -], or [ ] for a more recent presentation and further references).
The integrable sector of Einstein's equations is only a tiny fraction of their solution space. Unveiling the latter, in conjunction with its asymptotic symmetries and conserved charges, has been in the very early agenda of general relativity. It shares features with gauge theories because of general covariance, and has led Bondi to come out with his homonymous gauge, where a systematic resolution of Einstein's equations is possible as an expansion in powers of a radial coordinate. This delivers a set of functions of time and angular coordinates, obeying first-order time-evolution equations. For Ricci-flat spacetimes (asymptotically locally flat) this set is infinite, but it is finite for Einstein spacetimes with negative cosmological constant (asymptotically locally anti-de Sitter).
In modern language, the set of functions necessary for reconstructing the solution are said to be defined on the conformal boundary of the spacetime. In the asymptotically flat instance the conformal boundary is null infinity and features a Carrollian three-dimensional hypersurface. Bulk Einstein dynamics is therefore traded for boundary effective Carrollian conformal field dynamics. This statement is accurate when discussing Einstein's equations. Whether it could be promoted to a holographic principle akin to the better known AdS/CFT involving asymptotically anti-de Sitter spacetimes and conformal field theories defined on their time-like conformal boundary is a timely subject, currently under scrutiny.
How do hidden symmetries such as Ehlers' act on the Carrollian boundary data? This is the central question we would like to address in the present work. The conformal symmetries of the boundary reflect the asymptotic symmetries of the bulk. These define for instance the BMS 4 algebra (Bondi-van der Burg-Metzner-Sachs [ -]), which is isomorphic to the conformal Carroll algebra in three dimensions (3) (see [ , ]), and emerges upon appropriate fall-off conditions. From this perspective, wondering how the bulk hidden symmetries are embraced by the Carrollian boundary and what their interplay is with BMS 4 ≡ (3), is both natural and relevant. There is yet another motivation for pursuing this analysis. Following Geroch [ , ], the action of some Ehlers subgroup is a duality rotation in the plane of gravitational electric and gravitational magnetic charges, as are e.g. the mass and the nut charge. Ricci-flat spacetimes possess in fact multiple infinities of charges (not necessarily conserved), incarnated in pairs of electric and magnetic representatives, and originating from the infinitely many independent "subleading" degrees of freedom necessary for reconstructing the bulk solution, as well as the infinitely many generators of the asymptotic symmetry group. This picture has been widely conveyed through the work of Godazgar-Godazgar-Pope [ -] (see also [ -]) and amply deserves to be reconsidered in the light of hidden symmetries. The remarkable fact is here that such an analysis can be conducted exclusively on the boundary, where the charges are constructed (see e.g. [ ]) using the boundary dynamics combined with the three-dimensional-boundary Carrollian conformal isometries, the latter being always generated by the infinite-dimensional algebra BMS 4 ≡ (3, 1) + supertranslations [ ]. Translating the Ehlers group on the null boundary forcedly exhibits a mapping among the infinite towers of charges, which is obscured in a bulk approach. The boundary Carrollian geometry provides the most suitable language for clarifying these properties.
In the present work, we will analyse along the above lines the integrable sector of Ricci-flat spacetimes possessing a time-like Killing field, whose congruence coincides with the boundary Carrollian Other canonical gauges are Fefferman-Graham or Newman-Unti -see e.g. Refs. [ , ] for a review and more complete reading suggestions on this subject.
The original observation that triggered this "flat-holography" activity is described in Refs. [ , ]. A more systematic analysis in four dimensions was presented in [ ], which set the foundations for a Carrollian description of the dual theory, and provides a more complete reference list. Up-do-date developments in this vein are Refs. [ -]. field ξ. The latter has norm and twist -here , , . . . ∈ {0, . . . , 3}: where = √ − ( 0123 = 1). Assuming the spacetime be Ricci-flat, one shows that the one-form w = d is closed so locally exact, hence with a scalar function.
We define the three-dimensional space S as the quotient M /orb(ξ). This coset space is not a subspace of M unless ξ is hypersurface-orthogonal, which would imply zero twist with S the orthogonal hypersurface. A natural metric on Sis induced by g of M: which defines the projector onto Sas The fully antisymmetric tensor for ( . ) is = −1 √ − . Tensors of M, transverse and invariant with respect to ξ, are in one-to-one correspondence with tensors on S. If T is a tensor of S, the covariant derivative D defined following this correspondence,

( . )
with ∇ the Levi-Civita connection on (M, g), coincides with the Levi-Civita connection on (S, h). This sets a relationship between the Riemann tensor on Sand the Riemann tensor on M, generalizing thereby the Gauss-Codazzi equations to the instance where ξ is not-hypersurface orthogonal: (the calligraphic letters refer to curvature tensors of S). The Ricci-flat dynamics for is recast in the present framework in terms of This property actually holds more generally for Einstein spacetimes [ ].
With our conventions, this metric is definite-negative.
as well as and viewed as fields on S, packaged in and obeying the following equations: The first results from ( . ), while the second is obtained by a direct computation of the S-Laplacian acting on . HereD andR are the Levi-Civita covariant derivative and the Ricci tensor associated with the metrich displayed in ( . ).
Equations ( . ) feature two important properties. The first, due to Ehlers [ ], is the invariance under transformations maintainingh unaltered and mapping into This is the original instance where a hidden group, (2, R), reveals upon reduction with respect to an isometry. The second, described by Geroch in [ , ], is the method for reversing the reduction process, and finding a Ricci-flat four-dimensional spacetime with an isometry, starting from any solution of Eqs. ( . ) encoded in ′ + i ′ = ′ and ℎ ′ = 1 ′h .
To this end, one shows that the S-two-form defined as is closed. Thus, locally The one-form field η ′ , defined on S, can be promoted to a field on Mby adding the necessary exact piece such that its normalization be ′ = 1.
This defines a new Killing field on M ξ ′ = ′ η ′ ( . ) and the new four-dimensional metric reads: Equations ( . ) can be reached by varying a three-dimensional sigma-model action defined on S. This is at the heart of many developments about integrability and hidden symmetries -see the already quoted literature for more information.
The consequence of Möbius transformations on the Weyl tensor has been investigated in Ref.
Closing this executive reminder, we would like to add a remark. The (2, R) is hidden from the four-dimensional perspective, but explicit in the three-dimensional sigma-model, materialized here in Eqs. ( . ). Nevertheless, part of this group is in fact visible in four dimensions because it acts as fourdimensional diffeomorphisms; part is creating genuinely different Ricci-flat solutions. This can be illustrated in the concrete example of Schwarzschild-Taub-NUT solutions with mass and nut charge . The compact subgroup of rotations cos sin − sin cos ∈ (2) ⊂ (2, R) induces rotations of angle 2 in the parameter space ( , ), while non-compact transformations 0 1 / ∈ ⊂ (2, R) act homothetically, ( , ) → ( / , / ).

Ricci-flat spacetimes and Carrollian dynamics . Bulk reconstruction and resummable Ricci-flat metrics Choosing a covariant gauge
Four-dimensional Ricci-flat metrics are generally obtained as expansions in powers of a radial coordinate, in a designated gauge, usually Bondi or Newman-Unti. Appropriate fall-offs are assumed, and the solution is expressed in terms of an infinite set of functions of time and angles, obeying some evolution equations, mirroring Einstein's equations (see [ ] for details and further references). Can one define a three-dimensional boundary, and describe covariantly this set of functions and their dynamics?
The answer to this question has been known to be positive for a long time in the case of Einstein spacetimes. It is best formulated in the Fefferman-Graham gauge [ , ] -see also [ ] for a Weylcovariant extension of this gauge. The (conformal) boundary is a three-dimensional pseudo-Riemannian spacetime, and every order in the expansion brings a tensorial object with respect to the boundary geometry. All these are expressed in terms of two independent tensors: the first and second fundamental forms of the boundary, namely the boundary metric and the boundary energy-momentum tensor, which is covariantly conserved with respect to the associated Levi-Civita connection. This conservation translates those of Einstein's equations that have not been used in the process of taming the expansion.
The boundary covariance of the Fefferman-Graham gauge makes it elegant and suitable for holographic applications in the framework of anti-de Sitter/conformal-field-theory correspondence. Setting up a gauge that is covariant with respect to the boundary is therefore desirable as part of the effort to unravel a similar duality for asymptotically flat spacetimes. In this case, the conformal boundary is at null infinity and is endowed with a Carrollian geometry [ , ].
Carroll structures [ , , , -] consist of a + 1-dimensional manifold M = R × S equipped with a degenerate metric. The kernel of the metric is a vector field called field of observers. We will adopt coordinates ( , x) and a metric of the form ( . ) The coordinate system at hand is adapted to the space/time splitting. It is thus respected by Carrollian diffeomorphisms with Jacobian The clock form is dual to the field of observers with µ(υ) = −1: (Ω and depend on and x) and incorporates an Ehresmann connection, which is the background gauge field = d . Carrollian tensors depend on time and space x. We can summarize as follows the structure of the four-dimensional Ricci-flat solutions in the adver-A Carroll structure endowed with metric ( . ) and clock form ( . ) is naturally reached in the Carrollian limit ( → 0) of a pseudo-Riemannian spacetime M in Papapetrou-Randers gauge d 2 = − 2 Ωd − d 2 + d d , where all functions are -dependent with ≡ ( 0 = , x). It should be noticed here that the degenerate metric could generally have components along d , which would in turn give components to the field of observers. In this instance, the above Carrollian diffeomorphisms ( . ) play no privileged role, and plain general covariance is at work -without affecting the dynamics presented in App. B. This option is sometimes chosen (see e.g. [ ] for a general approach, or [ -] for an application to three-dimensional Minkowski spacetime), but it is always possible to single out the time direction supported by the fiber of the Carrollian structure, i.e. distinguish time and spatial sections with no conflict with general covariance. tised gauge, up to order 1 / 2 ( is four-dimensional Newton's constant): where the star designates a = 2 Carrollian Hodge duality as defined in Eq. (C. ). As anticipated, this expression is neither in Bondi gauge (no determinant condition -see [ , ]), nor in Newman- . Delving into the details of this gauge would bring us outside the main purpose of the present work. We will rather explain the various ingredients appearing in the above expression and insist on their Carrollian-covariant nature.
This includes the account of the required boundary data and the description of the evolution equations they obey so that the bulk metric be Ricci-flat.
All quantities entering expression ( . ) are defined on the conformal boundary and can be sorted as follows (see also the appendices for further information). Shear The dynamic shear is a symmetric and traceless Carrollian boundary tensor C ( , x) not to be confused with the geometric shear ( , x). It is a boundary emanation of the bulkcongruence shear, and is completely free, although it sources the evolution equations of other tensorial data. The dynamic shear carries information on the bulk gravitational radiation through the symmetric and traceless Bondi-like news:

Carrollian geometry
Referring to the complex coordinates introduced in App. C, we chose the orientation as inherited from the parent Riemannian spacetime: 0 ¯ = Ω √ 0 ¯ = iΩ 2 , where 0 = . In all quoted Eddington-Finkelstein type of gauges, is tangent to a null geodesic congruence. In Newman-Unti and in modified Newman-Unti this congruence is affinely parameterized, in contrast to Bondi. In modified Newman-Unti gauge, as opposed to the others, is not hypersurface-orthogonal. Indeed, the metric-dual form to is µ, which has a twist because of Ω and , the defining features of the gauge at hand: µ ∧ dµ = * d ∧ d ∧ µ (we have used Eqs. (A. ) and(C. )). In BMS gauge, one would set = 0, Ω = 1, and the round sphere. In Einstein spacetimes these two shears are proportional with the cosmological constant as a factor. In the asymptotically flat limit, the geometric shear is required to vanish, while the dynamic shear decouples.
With these definitions, the shear and the news are supported by genuine boundary conformal Carrollian-covariant tensors (weight −1 and 0), hence meeting the advertised expectations.

Carrollian fluid
The boundary Carrollian fluid of Ricci-flat spacetimes is the descendant of the relativistic boundary fluid in Einstein spacetimes in the vanishing speed of light limit, supported by the conserved energy-momentum tensor . It is described in terms of the energy density , the heat currents and , and the symmetric and traceless stress tensors Σ and Ξ [ , ]. The associated momenta of the fluid dynamics in the sense of App. B are as follows: As opposed to the relativistic boundary fluid, however, the Carrollian fluid is not free, but sourced by the shear, the news and the Carrollian Cotton descendants. Put differently, its dynamical equa- and (B. ) with (C. ). These data are related to the Bondi mass and angular momentum aspects, Notice that they do not exactly coincide with the original shear and news defined in BMS gauge. They vanish in Robinson-Trautmann spacetimes expressed in the gauge at hand, which is their defining gauge, although these solutions are radiating.
The presence of a non-vanishing energy flux Π = betrays the breaking of local Carroll boost invariance (see App. B, footnote ) in the boundary Carrollian dynamics associated with Ricci-flat spacetimes. This breaking accounts for bulk gravitational radiation, which in the boundary-covariant gauge designed here does not originate solely in the news ( . ) but is also encoded in the Carrollian energy flux Π = = 1 8 * and the Carrollian stressΠ = −Σ = − 1 8 * obeying Eq. (B. ) or equivalently (C. ). In Robinson-Trautman spacetimes e.g., the gravitational radiation is exclusively rooted in the latter Cotton descendants -see footnote and Ref. [ ].
We display for completeness these Carrollian equations, which coincide with Eqs. ( . ) and ( . ) of Ref.
Similarly to the expansion of Einstein spacetimes (in Fefferman-Graham or in the present gauge), fluid-related tensors appear at every order and not exclusively for 1 / , as expression ( . ) might suggest.
Further degrees of freedom Contrary to the asymptotically anti-de Sitter case, the above fluid data are not the only degrees of freedom besides the boundary geometry. An infinite number of Carrollian tensors are necessary to all orders in the radial expansion, as ( , x) in ( . ) at order 1 / , which obey Carrollian evolution -flux-balance -equations similar to those already displayed in footnote . These are dubbed "Chthonian" degrees of freedom.
We will not elaborate any further on the features of the expansion and the structure of the various evolution equations. The covariantization with respect to boundary Carroll diffeomorphisms and Weyl covariance is a powerful tool, rooted in the bulk general covariance. It can be supplemented with the boundary-fluid hydrodynamic-frame invariance at the expense of giving up radically the complete bulk gauge fixing. This requires a modified and incomplete Newman-Unti gauge, and has been performed for three bulk dimensions in Refs. [ -, ].

Resumming the series expansion
In certain circumstances the series ( . ) can be resummed. As advertised in the introduction, this occurs when conditions are imposed on the boundary data, which enforce specific features for the bulk Weyl tensor: . the dynamic shear C ( , x) should vanish, implying in particular the relation = 4 ; . all non-Carrollian-fluid related degrees of freedom should be discarded, as e.g.
( , x); . in ( . ) should be set to zero, which amounts to demanding the Carrollian momentum = be tuned with respect to a Carrollian Cotton descendant: In the configuration reached with the above conditions, the remaining degrees of freedom are those describing the boundary Carrollian geometry (metric, field of observers and Ehresmann connection), and the Carrollian-fluid energy density i.e. the Bondi mass aspect. Expression ( . ) is now resummed into an exact Ricci-flat spacetime of algebraically special type: The same equations are identically obeyed by the Carrollian Cotton tensors (C. ) and the geometric shear is vanishing. We are therefore left with two independent equations, which are (B. ) and (B. ): where and are given in geometric terms in (C. ) and (C. ), and is proportional to the Bondimass aspect, as stressed in item above. Equations ( . ) and ( . ) are those displayed in footnote with vanishing right-hand side.
From the above Eqs. ( . ) and ( . ) as well as Eq. (C. ) one can foresee that the energy density and the Carrollian Cotton scalar play dual roles. This will be formulated concretely in Sec. with reference to the boundary action of the Möbius group. Anticipating this argument, we introduce the following Carrollian complex scalarˆ ( , x) and vectorˆ ( , x): The aforementioned equations are thus recast as is remarkable that complicated equations as the latter can actually be tamed into a simple fluid conservation supplemented with a kind of self-duality requirement. It would have been unthinkable to reach such a conclusion without the null boundary analysis performed here and the corresponding Carrollian geometric tools. The latter provide definitely the natural language for unravelling asymptotically flat spacetimes.
A last comment before closing this section concerns the algebraic-special nature of the metric ( . ).
This is proven thanks to the Goldberg-Sachs theorem using the null, geodesic and, in the resummed instance, shear-free bulk congruence tangent to . The latter is part of the canonical null tetrad parallelly In complex celestial-sphere coordinates and¯ , see App. C, the null tetrad reads: with the usual relations k · l = −1, m ·m = 1 and d 2 res. Ricci-flat = −2kl + 2mm. Generically, k is a multiplicity-two principal null direction of the Weyl tensor, and using the tetrad at hand we find the following Weyl complex scalars: Unsurprisingly, all Ψs are spelled using the Carrollian descendants of the boundary Cotton tensor -as well as their derivatives in the higher-order terms.
Neither Ψ 3 nor Ψ 4 vanish in the instance of Petrov type D solutions, because l is not a principal null direction. Another tetrad is reached with a Lorentz transformation suitably adjusted for l ′ be a principal direction of multiplicity two whereas k ′ ∝ k, and Ψ ′ 3 = Ψ ′ 4 = 0.

. Bulk versus boundary isometries
The geometries under consideration possess at least one Killing vector field. A natural question to address concerns the boundary manifestation of a bulk isometry. At the same time such an analysis provides the recipe for designing bulk isometries from a purely boundary perspective. We will circumscribe our investigation to vector fields, which have no component along , and whose other components depend only on and x. We could be more general without much effort assuming e.g. an expansion in inverse powers of for the missing component and for the radial dependence of the others. However, this would unnecessarily sophisticate our presentation without shedding more light on our simple and robust conclusion: the bulk isometries at hand are mapped onto boundary Carrollian diffeomorphisms generated by strong Killing vectors (a summary on Carrollian isometries is available It is convenient for the subsequent developments to adopt bulk Cartan frame and coframe aligned with the boundary ( . ), (A. ) and ( . ): The components for the bulk metric ( . ) read (in order to avoid cluttering, we keep the "hat" on the time indices only, where potential ambiguity exists): Assuming a bulk vector of the form where ˆ = Ω − , we can determine the Lie derivative of the metric: Observe that everything is expressed in terms of boundary Carrollian geometric objects (see App. A).
Since the Killing components are -independent, the above Lie derivative vanishes if and only if the coefficients of every power of do. The independent conditions we reach for this to occur are

. Towers of charges and dual charges
The Carrollian dynamics emerging on the boundary as a consequence of bulk Einstein's equations, combined with the always available Carrollian conformal isometry group BMS 4 , enables us to define a variety of charges. These are not necessarily conserved, but even in that instance, their evolution properties are canonical and provide an alternative, tamed picture of the dynamics. Furthermore, they should ultimately pertain to those charges recently discovered and discussed from a bulk perspective [ -], based as usual on the asymptotic symmetries -also and unsurprisingly BMS 4 , under appropriate fall-off conditions. Making the precise contact with those works would require a translation of our findings into the This would bring us far from our goal, and we will limit ourselves to pointing out that the ten Newman-Penrose conserved charges vanish here because the spacetimes are algebraically special. These charges would have been otherwise associated with the = 1 "non-tilde" class introduced below, involving non-zero and in the non-algebraic instance. These charges are magnetic as the conservation of the Cotton is an identity valid off-shell. Furthermore, the Carrollian Cotton tensors are not exclusive to 1 / 2 : each order 1 / 2 +2 brings its share of off-shell

Ricci
Cot ( ) ,˜ Cot ( ) , and finally magnetic charges Cot ( ) , and˜ Cot ( ) , . Incidentally, it should be noticed that due to the relationships amongst the fluid and the Cotton (Eq. ( . ) in general plus Eq. ( . ) in the resummable family), the electric and the magnetic towers have a non-empty intersection:˜ ( ) , and˜ Cot ( ) , generally coincide.
We borrow here the phrasing electric and magnetic from Refs. [ , ].
Let us for concreteness overview the situation in the resummable instance, Eq. ( . ). Expanding the resummed factor 1 / 2 , we find the following results.
Regarding the charges and their evolution, always conserved. These charges are purely geometric because they are integrals over S which do not involve the energy density , as opposed to (0 The latter are conserved for strong Carrollian Killings. Other charges might also be conserved for specific Carrollian conformal Killings, or depending on the configuration.
Cot ( ) , Cot ( ) ,˜ Cot ( ) and˜ Cot ( ) built out of the leading = 0: Their divergences (B. ) read: These determine the evolution (B. ) of the charges (B. ), from which we learn that˜ Cot (0) are also conserved off-shell, as other magnetic charges are in specific situations.
Several comments are in order here concerning the above sets of charges obtained for the resummable metrics ( . ). The tower of electric geometric charges˜ ( ) , , constructed upon multiplying the integrand of ( . ) by * 2 , coincides with its magnetic counterpart˜ Cot ( ) , obtained likewise using ( . ). In = 2, if are the spatial components of a conformal Killing field, so are * . Hence the set of all s is identical to that of * s. The associated charges could be called "self-dual, " and in total three distinct towers emerge: the self-dual ˜ ( ) , ≡ ˜ Cot ( ) , , the electric Among the above towers of BMS 4 charges, always present but not always conserved, one finds those corresponding to the bulk isometries, whenever present. Indeed, as discussed in Sec. . , bulk Killings of the form ( . ) are associated with boundary strong Carrollian Killing vector fields. Combined as previously with the leading and subleading, electric and magnetic momenta, they generate two electric and two magnetic towers of charges: (0) are always conserved, but part of them may be trivial or not independent. The subleading are neither necessarily conserved, nor always independent, and have the status of electric and magnetic multipole moments. .

Time-independent solutions Reconstruction from the boundary
In view of the forthcoming Ehlers-Geroch reduction, we will now assume the existence of a time-like With the present choice, none of the Carrollian building blocks of d 2 res. Ricci-flat depends on . As a consequence (see Apps. A and C) = 0 and = ln Ω. The latter can be set to zero with a timeindependent Weyl rescaling, which therefore amounts to setting Ω = 1. This is an innocuous gauge fixing that will be assumed here because it allows to severely simplify the dynamics. Backed with time independence, Carrollian Weyl-covariant derivatives become ordinary Levi-Civita derivatives, and the only non-vanishing tensors are the following, in complex coordinates with = ( ,¯ ) -see App. C: Their Weyl components are given in Eq. ( . ) -see also footnote .
where Δ = 2 2 ¯ . To these one should add the energy density (i.e. the Bondi mass aspect) , as well as another scalar which is 1 2 ∇ and should not be confused with the two-form ̟ = 1 2 d ∧ d , i.e. the Hodgedual of the scalar * = − 1 2 ∇ * displayed explicitly in ( . ). These two real twist scalars are adroitly combined into the complex Carrollian twistˆ The equations of motion ( . ), ( . ) (or in the form ( . ), ( . ) withˆ defined in ( . )) are recast as The first shows that the curvature is required to be a harmonic function i.e.
and althoughˆ ( ) is an arbitrary holomorphic function, the freedom is rather limited as must also be the Laplacian of ln . Besides the constant-curvature cases, one solution has been exhibited thus far [ ] (up to holomorphic coordinate transformations): = −3( +¯ ) realized with = ( +¯ ) 3 /2 . We will not specify any particular choice for the moment. For future use, we define the imaginary part ofˆ ( ) as another harmonic function From Eqs. ( . ) and ( . ), we infer that − is the real part of an arbitrary holomorphic functionˆ ( ), whereas the imaginary part of the latter is 8 ; both are harmonic functions. Given and , we can proceed with Eq. ( . ) and find * , from which it is always possible to determine and ¯ .
Although the focus of the present work is not to solve Einstein's equations, we will elaborate for illustrative purposes on the steps we've just described, without delving into fine questions like completeness or gauge redundancy of the solutions. Note in passing how remarkably the Carrollian boundary formalism is adapted to the framework of Ricci-flat spacetimes, allowing to convey often complicated expressions in a very elegant manner, and sorting naturally otherwise scattered classes of solutions (the ones we present can be found in various chapters of Refs. [ , ]). Several distinct instances appear, which require a separate treatment.
Non-constant This is the generic situation, although in practice the most obscure regarding the interpretation of the bulk geometries. As already mentioned, very few s are expected to possess a non-constant harmonic curvature , but assuming one has one, accompanied by its holomorphic functionˆ ( ), and making a choice for the arbitrary holomorphic functionˆ ( ), Eq. ( . ) can be solved for * , which is expressed using ( . ) with Ehresmann connection Constant This implies thatˆ ( ) is also constant and the above solution is invalid. The situation at hand is the most common, however, as it captures three standard instances: spherical, flat or hyperbolic foliations. We can parameterize the function as follows: with , arbitrary real constants and an arbitrary complex constant, leading to Several cases emerge, which must be treated separately.
is an arbitrary (possibly constant) harmonic function, and Eq.
with ( ) an arbitrary holomorphic function. It is reached with the following Ehresmann connection (ˆ 0 is a real constant): and an arbitrary holomorphic function ( ), we find * ( and (ˆ 0 is a real integration constant) The last two cases have in common the instance where = = 0, realized with vanishingˆ and constant .
As already noticed, all solutions described in a unified fashion here can be found in the earlier quoted literature under distinct labels. Discussing them would take us outside of our objectives. We will only emphasize a notorious subclass, which is the Kerr-Taub-NUT family. For the latter, the curvature is constant ( . ) and realized e.g. with = 0. Two distinct instances emerge: vanishing and non-vanishing , respectively obtained with vanishing and non-vanishing .
• For = 0 (i.e. = constant), we use Eqs. ( . ) and ( . ) withˆ 0 = 2 , It should be stressed that part of the present solution space originates in gauge freedom. In particular, ( ,¯ ) being Weylcovariant of weight 3 (see App. C), it can always be reabsorbed by a boundary Weyl transformation, which is in turn neutralized by a bulk -rescaling. Such a boundary transformation will bring Ω back with non-vanishing , which we have set to zero, and this is the reason we cannot here restrict to constant andˆ .
Both for vanishing and non-vanishing ,ˆ 0 has been tuned to ensure that does not appear in , displayed in ( . ) and ( . ). There is no principle behind this choice, it is simply in line with standard conventions for the Kerr-Taub-NUT family. As a consequence, defined in ( . ) vanishes. This leads to Observe the absence of nut charge in the present case.

A remark on the rigidity theorem
The rigidity theorem asserts that under appropriate hypotheses, the isometry group of stationary asymptotically flat spacetimes contains R × (1). This theorem is best presented in Refs. [ , ], where the necessary assumptions are stated more accurately than in the original discussions (see e.g. [ ]). Our framework does embrace stationary spacetimes. However, we have been agnostic regarding analyticity or regularity properties, which turn out to be fundamental for the applicability of the theorem at hand.
Hence, we have no reason to foresee any additional (1) symmetry in all reconstructed solutions of the present chapter.
Aside from mathematical rigor, we can recast the conceivable disruption of the rigidity theorem from the boundary perspective, which has been our viewpoint. We have shown in Sec. . that a bulk Killing field is mapped onto a Carrollian strong Killing on the boundary. The generator of the desired (1) is of the form ( . ) with no time leg i.e. = 0, and no time dependence in as imposed by ( . ): We could keep non-vanishing and perform a more thorough analysis. This would not alter the conclusions, which are meant here to illustrate possible boundary faults in the rigidity theorem.
solutions to the equations ( . ):

Charge analysis
We would like to close the present section with a brief account on the charges of the Ricci-flat solutions under investigation. Gravitational charges disclose the identity of a background and, as we have proposed in Sec. . , boundary Carrollian geometry supplies alternative techniques for their determination and the study of their conservation. These techniques are still in an incipient stage though, because the contact with the standard methods still needs to be elaborated. Furthermore, non-radiating configurations, in particular stationary and algebraically special, offer a limited playground in this programme.
We would like nevertheless to summarize the situation, in view of the follow-up discussion on Möbius hidden-group action, Sec. . .
The simplest non-vanishing charge is the electric curvature defined in (C. ): Divided by the volume of S, this is simply the average Gauss curvature. Note in passing that the charges defined here are extensive, hence the integrals may reveal convergence issues, in particular when S is non-compact. Normalizing with Vol = ∫ S d ∧d¯ i 2 is the simplest way to fix this divergence. Alternatively, S could be compactified -quotiented by a discrete isometry group. We will leave this discussion aside, as it would be better addressed within attempts to make sense of Ricci-flat black holes with non- Remember that here = 0, and the geometry is -independent with vanishing , ,Â,R as well as . The integrals can be performed by setting = e , where 0 ≤ < 2 and = √ 2 tan 2 , 0 < < for Indeed, the Carrollian conformal Killings used in expressions ( . ) and ( . ) are (C. ) with (see (C. )). Observe also that˜ =˜ Cot = 0 so thatK ( ) =K Cot ( ) = 0. Generically, K ( ) and K Cot ( ) are non-zero though, because the conformal Killing vectors are not necessarily strong and due to the time dependence, here encoded exclusively in their component ˆ . The corresponding charges are ultimately expressed as integrals of combinations ofˆ ,ˆ ,ˆ ,ˆ , * , , and of their derivatives.
For concreteness, we will illustrate the above with the distinctive strong Carrollian conformal Killing field , i.e. the generator of the Ehlers-Geroch bulk three-dimensional reduction. For this Killing field, the "tilde" Carrollian charges vanish. In example, for the leading charges ( = 0 in the coding of Sec. . ), up to boundary terms with respect to ( . ) and ( . ) (and a factor −8 for the former), handily combined into The indices stand for magnetic and electric masses. These mass definitions carry some arbitrariness since, as a consequence of time independence, each of the terms in the integrals provide a separate welldefined charge. We will turn back to this when discussing the action of the Möbius group, in Sec. . . ( . )) are combined in the complex higher-angular-momentum multipole moments Although the components of the Ehresmann connection enter the expression of the Carrollian charges (B. ), upon integration by parts, they are traded for * or .
Using ( . ) and ( . ) with ˆ = 1 and = 0, we find = − , = − 1 8 * , Cot = − and Cot = − . with = 1 + 1 2 ¯ , which are non-zero if one rotation parameter or is present. We find for example: In order to proceed, we are called to follow the steps for the Geroch reduction described in Sec. , i.e. determine as defined in ( . ) with norm For the twist we use Eq. ( . ), expressed as where "★" stands for the four-dimensional Hodge duality. The latter one-form is exact on-shell and we find the following potential (Eq. ( . )): On-shellness is implemented here through boundary dynamics as summarized in Sec. . , i.e. in Eqs.
The premier Ehlers transformation rules are ( . ) and the invariance ofh . From these follows the rest of the construction, i.e. the transformation of ℎ and the oxidation toward ′ . In the present framework, we have to some extent locked the gauge, via the resummed bulk expression ( . ). Ehlers transformations are not designed a priori to maintain this form, and they are generally expected to require further coordinate transformations. It is rather remarkable that, to this end, a local (i.e. celestialsphere dependent) shift in the radial coordinate suffices.
Using for convenience holomorphic and antiholomorphic coordinates as introduced in App. C, expression ( . ) is recast as follows: Combining ( . ) with ( . ), we obtain the following boundary transformations: and plus the radial shift These transformation rules leave indeed ( . ) invariant. As advertised earlier, they are local, providing a direct transformation ( . ) of the boundary metric. The transformation of the energy density is obtained from ( . ) using ( . ): The transformation of is inferred similarly: finally leading to ( . ), which assumes the form ( . ) primed. The new bulk Killing vector ξ ′ = ′ η ′ is again .
One could alternatively adopt a new radial coordinate defined as˜ = + that is invariant under Möbius transformations. This is actually mandatory in order to reach boundary (2, R)-covariant tensors from the bulk, as we will discuss in Sec. . . It furthermore coincides with the radial coordinate of Ref. [ ] § provided 0 = − (origin of the affine parameter along the geodesic congruence tangent to -see footnote ).
In the example of the Kerr-Taub-NUT family treated at the end of Sec. . , the specific choices of = 1 2 ¯ + 1, = 1 and * = 0 (this was not explicitly demanded) are stable only under cos sin − sin cos ∈ (2, R). For this transformation, using ( . ) we find ′ + i ′ = ( + i )e −2i . Observe that ( . ) will switch on a non-zero though, as opposed to its original value in the family at hand (see footnote ).

. Charges and
(2, R) multiplets Carrollian charges have been introduced in Sec. . and further discussed for stationary and algebraic spacetimes in Sec. . . Two generic charges were found and displayed in ( . ) and ( . ). The former is purely geometric and stands for the integrated curvature of the celestial sphere; the latter carries genuine dynamic information captured in the electric and magnetic masses. It is legitimate to wonder how these quantities behave under Möbius transformations, and possibly tame them in (2, R) multiplets.
Although ideally this programme should be conducted for reductions along generic bulk Killing fields and no special algebraic structure -these would be non-resummable, i.e. of the form ( . ), and labelled by a possibly plethoric set of independent charges -we will pursue it here for illustrative purposes in the restricted framework at hand.
The curvature charge ec in ( . ) is invariant under Ehlers' (2, R), and this is inferred using the transformation laws ( . ) and ( . ). The mass charge m , Eq. ( . ), is not, but its transformation (see ( . ), ( . ) and ( . )) suggests that it might belong to some (2, R) multiplet or, more accurately, that it may be modified to this end -we have this freedom owing to time independence. Actually, a slight amendment to the charge m , namely is (2, R)-invariant. We can even go further and apply the following pattern to generate (2, R) triplets. Suppose we identify a Carrollian two-form transforming under (2, R) as This allows to design an (2, R) two-form triplet, i.e. a symmetric rank-two tensor, transforming as where The same holds for the complex-conjugate triplet:¯ 1 =¯ ,¯ 2 = −iˆ ¯ and¯ 3 = −ˆ 2¯ . An (2, R) triplet of charges is thus reached as and ≡ 1 3 − 2 2 is invariant under Möbius transformations. The above strategy can be readily applied. Two-forms transforming as in ( . ) can be found, inspired by the structures of the charge ( . ) and of the Carrollian currents ( . ) and ( . ), given the expressions of the Carrollian twist ( . ), the Carrollian curvature ( . ), and the Carrollian Cotton tensors ( . ), ( . ), ( . ) and ( . ). We here exhibit two such Carrollian forms: These lead along ( . ) to two triplets of charges, which do not carry more information than the original where "★ 3 h " stands for the three-dimensional Hodge-dual on Sequipped withh displayed in ( . ). It is remarkable that the asymptotic limit of this two-form triplet coincides with those designed earlier from Carrollian boundary considerations. This statement is captured in the following result: where˜ = + was introduced in footnote as an (2, R)-invariant radial coordinate, which must be used here in order to guarantee that the limit preserves the (2, R) behaviour.

Conclusions
When a four-dimensional spacetime geometry is invariant under the action of a one-dimensional group of motions, a reduction can be performed and vacuum Einstein dynamics reveals a symmetry under Möbius transformations. Our main motivation was to exhibit this action from a holographic perspective, namely on the three-dimensional boundary of the Ricci-flat configuration at hand. We have successfully bulk spacetimes and pseudo-Riemannian boundaries do provide a conserved Cotton tensor, which contracted with any boundary conformal Killing vector leads to a conserved current, hence a conserved charge. This powerful tool is undermined by the limited -if any -number of conformal isometries on arbitrary three-dimensional Riemannian spacetimes. The remarkable spin-off about Carrollian boundaries is the existence of an infinite-dimensional conformal group, which makes this method of charge determination a serious alternative to the more standard bulk asymptotic techniques. Following the Cotton pattern, electric towers of charges are constructed with the fluid dynamical data, which can only enjoy on-shell conservation -the same would hold in AdS boundaries with the aforementioned limitation. On both electric and magnetic sides, the towers of charges are multiplied ad nauseam, beyond their leading components.
Our present investigation on towers of charges designed from a boundary standpoint is radically novel and deserves a systematic extension. It has been here confined in the integrable case, where the infinite set of observables is redundant and shrinks to the elementary "leading" data -our tentative definition of subleading currents might have turned too naive, hadn't it reproduced successfully the multipole moments. Moreover, our main goal being primarily on boundary Ehlers action, we have assumed a time-like bulk isometry, which further reduces this set. Besides, the chosen time-like Killing field was aligned with the fibre of the boundary Carrollian structure, which screens the black-hole acceleration parameter and avoids exploring head-on the uncharted subject of Carrollian reductions. The latter is the mathematical tool to be developed for unravelling the bulk-to-boundary relationship of hidden symmetries in Ricci-flat spacetimes. It could encompass bulk reductions along space-like isometries, which are interesting because they leave room for gravitational radiation, probing the interplay between Ehlers Möbius group, time evolution and charge non-conservation. Last, we did not address the question of the charge algebra and its potential central extensions, or discussed other more general related physical aspects. All this calls for a thorough comparison to alternative approaches such as those of Refs. [ -] based on Newman-Penrose formalism -or to applications [ , -, ].
In the chapter of charges, in spite of the various limitations just stated, we have successfully described the Ehlers Möbius action, and discussed the organisation of available charges in (2, R) multiplets. This enabled us to recover from a Carrollian viewpoint the triplet of Komar charges inferred by Geroch in its original publication [ ]. Again, this result should be considered as a first step toward a methodical (2, R) taming of the above towers of electric/magnetic currents and charges in more general situations. These objects should include the boundary attributes of the bulk Weyl tensor, whose behaviour under Möbius transformations has been addressed in [ ].
The importance of the boundary covariantization -Carroll and Weyl -is yet another feature we would like to stress, as it hasn't been sufficiently appreciated in the literature. This characteristic is absent from Bondi or Newman-Unti gauges, where the formalism might suggest that the relevant part of the conformal boundary is its two-dimensional spatial section -the celestial sphere. We heavily insist on the three-dimensional and Carrollian nature of the boundary, which is made manifest in the gauge The Petrov-algebraic spacetimes ( . ) accommodate axisymmetric time-dependent solutions of the Robinson-Trautman type, whose final state is the C-metric -see. [ ] § . . we have been using. In ordinary AdS/CFT holography the Fefferman-Graham gauge is superior for this reason. One should likewise use a truly boundary-covariant gauge in flat holography and take advantage of it, as we modestly did for exhibiting the action of Ehlers' group, or for discussing the charges and their conservation. No boundary approach of this sort would have been possible in the more conventional gauges. Correspondingly, flat holography based on a purely celestial gauge is bound to be incomplete.
It is worth mentioning that Ehlers' (2, R) group is the first and simplest example of a hidden symmetry. As pointed out in the introduction (see the references proposed there), more involved reductions reveal richer symmetries and the underlying dynamics is captured by elegant sigma models in various dimensions. Recasting this knowledge in a holographic fashion, we could possibly learn more, or at least differently, not only about hidden symmetries but also on flat holography. Carrollian reductions might again be the appropriate tool.
On a more speculative tone, our results suggest that a boundary analysis might reveal more general or unexpected duality properties. The paradigm of anti-de Sitter spacetimes, where the (2, R) is broken in the bulk but restored on the boundary, calls for a systematic investigation that would complement the heuristic discussion of Ref. [ ], and possibly uncover novel instances of boundary duality symmetries, associated e.g. with an asymptotic Killing field rather than a plain reduction along Killing orbits. One could even be more audacious and entertain the idea of a "boundary" analysis for half-flat spaces (this is vaguely motivated by footnote ), which have attracted some attention in relation with

A Carrollian covariance in arbitrary dimension
Carroll structures on M = R × S were introduced in Sec. . with emphasis on the covariance properties they enjoy when the time coordinate is aligned with the fiber of the structure. In the present appendix we will elaborate on this subject, treating in particular Carrollian covariant and Weyl-covariant derivatives.
The Carrollian transformations (Eqs. ( . ) and ( . )) are connection-like (non-covariant) for and , and density-like for and Ω: The vector fields dual to the forms d areˆ and transform covariantly under ( . ) together with the metric ( . ), and the fields ( . ) and ( . ): The vectorsˆ and υ do not commute. They define the Carrollian vorticity and acceleration: similarly appearing in A Carroll structure is also equipped with a metric-compatible and field-of-observers-compatible connection (strong definition). Due to the degeneration of the metric, such a connection is not unique, but it can be chosen as the connection inherited from the parent relativistic spacetime (see footnote ), obeyingˆ [ ] = 0,∇ = 0 and leading to the Levi-Civita-Carroll spatial covariant derivative∇ . The ordinary time-derivative operator 1 Ω acts covariantly on Carrollian tensors. However, it is not metric-compatible because depend on time and a temporal covariant derivative is defined requiring To this end, we introduce a temporal connection (a sort of extrinsic curvature of the spatial section S) which is a symmetric Carrollian tensor spliting into the geometric Carrollian shear (traceless) and the Carrollian expansion (trace). The action ofˆ on any tensor is obtained using Leibniz rule plus the action on scalars and vectors: The commutators of Carrollian covariant derivatives define Carrollian curvature tensors: whereˆ is the Riemann-Carroll tensor. The Ricci-Carroll tensor and the Carroll scalar curvature are Similarly, space and time derivatives do not commute: further Carrollian curvature tensors.
The boundary geometry -be it pseudo-Riemannian or Carrollian -enjoy conformal properties.
Weyl transformations are defined through their action on elementary geometric data The Weyl-Carroll space and time covariant derivatives are metric-compatible. For a scalar function Φ and a vector of weight , we find: The weights are not altered by the spatial derivative andD = 0. One also defines both are of weight + 1. FurthermoreD = 0, using Leibniz rule.
We finally obtain and =ˆ −ˆ , are weight-0 Weyl-covariant tensors. Tracing them we obtain: of weights zero and 2. The Weyl-covariant Carroll-Ricci tensor is not symmetric,R [ ] = − 2 Ω , and a weight-1 curvature form also appears with

B Conformal Carrollian dynamics and charges
A complete account on the subject of dynamics and charges with the present conventions is available in Refs. [ , ]. We summarize here the necessary items, in particular regarding the Weyl-covariant side, which is relevant on the holographic boundaries.
The basics are encoded into four Carrollian momenta, replacing the relativistic energy-momentum tensor, which are obtained by varying some (effective) action with respect to , and Ω (the fourth momentum is not necessarily obtained in this way -for details see [ ]). These are the energy-stress tensor Π , the energy flux Π , the energy density Π as well as the momentum , of conformal weights + 3, + 2, + 1 and + 2. Extra momenta can also emerge as more degrees of freedom may be present. This phenomenon occurs when studying the small-limit of a relativistic energy-momentum tensor and the corresponding conservation equations. Keeping things rather minimal with onlyΠ the equations read: as a consequence of the assumed Weyl invariance.
Equations (B. ), (B. ), (B. ) and (B. ) are the Carrollian emanation of the relativistic conservation equation ∇ = 0. As for the relativistic instance, conformal isometries lead to conserved currents and conserved charges. Let ξ be a + 1-dimensional vector Using the language of fluids, Π appears as the zero-limit of the relativistic energy density, Π and are the orders one and 2 of the relativistic heat current, whereasΠ and Π are the orders 1 / 2 and one of the relativistic stress. A non-vanishing Carrollian energy flux Π breaks local Carroll-boost invariance (see e.g. [ ]) and makes its dual variable i.e. the Ehresmann connection = d dynamical. This is neither a surprise nor a caveat. On the one hand, Carrollian dynamics, i.e. dynamics on geometries equipped with a degenerate metric, is often reached as a vanishing-limit of relativistic dynamics and naturally breaks local Carroll boosts, even when the original relativistic theory is Lorentz-boost invariant. Indeed, invariance under local Lorentz boosts sets symmetry constraints on the components of the relativistic energy-momentum tensor, but not on their behaviour with respect to 2 , leaving the possibility of persisting energy flux Π and "over-stress"Π related through Eq. (B. ). A similar phenomenon occurs in Galilean theories, defined on spacetimes with a degenerate cometric, where the Galilean momentum is possibly responsible for the breaking of local Galilean-boost invariance. On the other hand, it is fortunate that this happens in the present instance (one of the very few known applications of Carrollian dynamics), when passing from the relativistic boundary of asymptotically anti-de Sitter spacetimes to the Carrollian boundary of their asymptotically flat relatives, as the Carrollian energy flux accounts for non-conservation properties resulting from bulk gravitational radiation, whereas the Ehresmann connection is part of the Ricci-flat solution space.
The Lie derivative along ξ = ˆ 1 Ω + ˆ of a general Carrollian tensor reads: with ν = d and Due to the degeneration of the metric on M, the variation of the field of observers υ is not identical to that of the clock form µ.
Isometries are generated by Killing fields of the Carrollian type (B. ), required to obey [ , , , ]: The clock form is not required to be invariant. Carrollian conformal Killing fields must satisfy This set of partial differential equations is insufficient for defining conformal Killing fields. One usually imposes to tune versus (see [ , , ] for a detailed presentation) so that the scaling of the metric be twice that of the field of observers: (the conformal weight of ˆ is −1, that of is zero). Again, the clock form is not involved. If one demands the latter be invariant under the action of a Killing field, or aligned with itself under the action of a conformal Killing, which in both cases amounts to setting (this is a conformal rewriting of given in (B. )), then the corresponding (conformal) isometry generator will be referred to as (conformal) strong Killing vector field.
The charge associated with the current ( , ) is an integral at fixed over the basis S and obeys the following time evolution: The last term is of boundary type with * the S-Hodge dual of d . Generally, one can ignore it owing to adequate fall-off or boundary conditions on the fields.
Suppose that ξ is the generator (B. ) of a Carrollian diffeomorphism. It can be used to create two currents out of Π ,Π , Π , and Π [ , ]: We stress here that if more momenta were present, more currents would be available. Notice that these are the specific conformal weights ensuring the Carroll divergence in (B. ) be identical to the Weyl-Carroll divergence. For , of general weights ( , + 1) we find instead 1 Two charges can be defined following (B. ): ˜ and . The former is conserved, whereas the latter isn't for generic isometries unless the field configuration has vanishing energy flux Π , i.e. if local Carrollboost invariance is unbroken. The breaking of local Carroll-boost invariance hence appears as the trigger of non-conservation laws. This peculiarity was risen in [ , ] and further illustrated with concrete field realizations in [ ]. In four-dimensional Ricci-flat spacetimes, this boundary non-conservation is the consequence of bulk gravitational radiation, as mentioned previously in footnote . Observe nevertheless that irrespective of the energy flux Π , the (conformal) strong Killings introduced earlier do lead to full conservation properties as a consequence of (B. ).  where we have introduced two weight-2 Weyl-covariant scalar Gauss-Carroll curvatures: ) and * = 1 2 . These obey Carroll-Bianchi identities:

C Three dimensions and the Carrollian Cotton tensor
Thanks to the identities (C. ) and (C. ), the couples K , −R −D and Â , − * R of weights Upon regular behaviour, the boundary terms vanish and the curvature charges are both conserved.
Besides the various curvature tensors, which are second derivatives of the metric and the Ehresmann connection, one defines third-derivative tensors, the descendants of the relativistic Cotton tensor. We will here limit our presentation to the instance = 0, which is the appropriate framework when solving Einstein's equations in the bulk. This reduces the number of tensors to five, a weight-3 scalar, two weight-2 forms and two weight-1 two-index symmetric and traceless tensors: As a consequence of the relativistic conservation of the Cotton tensor, its Carrollian descendants obey When the geometric Carrollian shear vanishes, the time dependence in the metric is factorized as . One then shows [ , ] that the Carrollian conformal isometry group is the semi-direct product of the conformal group of¯ (x) with the infinite-dimensional supertranslation group. The former is generated by (x), the latter by (x), and the Carrollian conformal Killing fields read: This result is valid in any dimension. At = 2,¯ (x) is conformally flat and (x) generate (3, 1). shear displayed in (A. ) vanishes. We will here assume that this holds and present a number of useful formulas for Carrollian and conformal Carrollian geometry. These geometries carry two further pieces of data: Ω( , ,¯ ) and

The conservation of the
with ¯ ( , ,¯ ) =¯ ( , ,¯ ). Our choice of orientation is inherited from the one adopted for the relativistic boundary (see footnote ) with The first-derivative Carrollian tensors are the acceleration (A. ), the expansion (A. ) and the scalar vorticity (A. ), (C. ): Curvature scalars and vector are second-derivative (see (C. ), (A. )): and we also quote: Regarding conformal Carrollian tensors we remind the weight-2 curvature scalars (C. ): This amounts to setting √ = i / 2 in coordinate frame and ¯ = −1. The volume form reads d 2 √ = d ∧d¯ i 2 . We also quote for completeness (useful e.g. in Eq. (C. )): ¯ Ω 2 ln with = 2 2 ¯ ln the ordinary Gaussian curvature of the two-dimensional metric (C. ). and the weight-1 curvature one-form (A. ): The three-derivative Cotton descendants displayed in (C. )-(C. ) are a scalar of weight 3 ( * is of weght 1), two vectors of weight 2, and two symmetric and traceless tensors of weight 1. Notice that in holomorphic coordinates a symmetric and traceless tensor has only diagonal entries: ¯ = 0 = ¯ . We also remind for convenience some expressions for the determination of Weyl-Carroll covariant derivatives. If Φ is a weight-scalar function For weight-form components and ¯ the Weyl-Carroll derivatives read: while the Carrollian covariant derivatives are simply: Finally, Using complex coordinates, we can recast the conformal Killing vectors of a shear-free Carrollian spacetime M in three dimensions, given in Eqs. (C. ) and (C. ). These are expressed in terms of an arbitrary real function ( ,¯ ), which encodes the supertranslations, and the conformal Killing vectors of flat space dl 2 = 2d d¯ . The latter are of the form ( ) + ¯ (¯ ) ¯ , reached with any combination of ℓ +l or i ℓ −l , where obeying the Witt ⊕ Witt algebra: ) and referred to as superrotations. Usually one restricts to (3, 1), generated by = 0, ±1. The conformal Killing fields of M are thus The structure (3, 1) + supertranslations -or (Witt ⊕ Witt) + supertranslations -is recovered in