Flat holography and Carrollian fluids
HJE
Flat holography and Carrollian fluids
Luca Ciambelli 0 1 3 6 7
Charles Marteau 0 1 3 6 7
Anastasios C. Petkou 0 1 3 4 5 7
P. Marios Petropoulos 0 1 3 6 7
Konstantinos Siampos 0 1 2 3 5 7
0 Geneva 23 , 1211 Switzerland
1 Thessaloniki , 54124 Greece
2 Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics
3 Universit ́e ParisSaclay , Palaiseau, Cedex 91128 France
4 Department of Physics, Institute of Theoretical Physics, Aristotle University of Thessaloniki
5 Theoretical Physics Department , CERN
6 CPHT  Centre de Physique Th ́eorique, Ecole Polytechnique , CNRS UMR 7644
7 University of Bern , Sidlerstrasse 5, Bern, 3012 Switzerland

locally flat spacetimes is reached smoothly from the zerocosmologicalconstant limit of
antide Sitter holography. To this end, we use the derivative expansion of fluid/gravity
correspondence. From the boundary perspective, the vanishing of the bulk cosmological
constant appears as the zero velocity of light limit. This sets how Carrollian geometry
emerges in flat holography. The new boundary data are a twodimensional spatial
surface, identified with the null infinity of the bulk Ricciflat spacetime, accompanied with a
Carrollian time and equipped with a Carrollian structure, plus the dynamical observables
of a conformal Carrollian fluid. These are the energy, the viscous stress tensors and the
heat currents, whereas the Carrollian geometry is gathered by a twodimensional spatial
in a closed form in EddingtonFinkelstein gauge under further integrability conditions
inherited from the ancestor antide Sitter setup. These conditions are hinged on a duality
relationship among fluid friction tensors and Cottonlike geometric data. We illustrate
these results in the case of conformal Carrollian perfect fluids and RobinsonTrautman
viscous hydrodynamics. The former are dual to the asymptotically flat KerrTaubNUT
family, while the latter leads to the homonymous class of algebraically special Ricciflat
spacetimes.
gravity correspondence
ArXiv ePrint: 1802.06809
1 Introduction
2 Fluid/gravity in asymptotically locally AdS spacetimes
2.1
The derivative expansion
2.2 The resummation of AdS spacetimes
3 The Ricciflat limit I: Carrollian geometry and Carrollian fluids
4 The Ricciflat limit II: derivative expansion and resummation
Back to the derivative expansion
Resummation of the Ricciflat derivative expansion
5 Examples
Stationary Carrollian perfect fluids and Ricciflat KerrTaubNUT families
Vorticityfree Carrollian fluid and the Ricciflat RobinsonTrautman
6 Conclusions A Carrollian boundary geometry in holomorphic coordinates 1 5
Sitter bulk spacetimes. Since the genuine microscopic correspondence based on type IIB
string and maximally supersymmetric YangMills theory is deeply rooted in the antide
Sitter background, phenomenological extensions such as fluid/gravity correspondence have
been considered as more promising for reaching a flat spacetime generalization.
The mathematical foundations of holography are based on the existence of the
FeffermanGraham expansion for asymptotically antide Sitter Einstein spaces [1, 2]. Indeed, on
the one hand, putting an asymptotically antide Sitter Einstein metric in the
FeffermanGraham gauge allows to extract the two independent boundary data i.e. the boundary
metric and the conserved boundary conformal energymomentum tensor. On the other
hand, given a pair of suitable boundary data the FeffermanGraham expansion makes it
possible to reconstruct, order by order, an Einstein space.
More recently, fluid/gravity correspondence has provided an alternative to
FeffermanGraham, known as derivative expansion [3–6]. It is inspired from the fluid derivative
– 1 –
expansion (see e.g. [7, 8]), and is implemented in EddingtonFinkelstein coordinates. The
metric of an Einstein spacetime is expanded in a lightlike direction and the information
on the boundary fluid is made available in a slightly different manner, involving explicitly
a velocity field whose derivatives set the order of the expansion. Conversely, the boundary
fluid data, including the fluid’s congruence, allow to reconstruct an exact bulk Einstein
Although less robust mathematically, the derivative expansion has several advantages
over FeffermanGraham. Firstly, under some particular conditions it can be resummed
leading to algebraically special Einstein spacetimes in a closed form [
9–14
]. Such a
resummation is very unlikely, if at all possible, in the context of FeffermanGraham. Secondly,
boundary geometrical terms appear packaged at specific orders in the derivative
expansion, which is performed in EddingtonFinkelstein gauge. These terms feature precisely
whether the bulk is asymptotically globally or locally antide Sitter. Thirdly, and contrary
to FeffermanGraham again, the derivative expansion admits a consistent limit of vanishing
scalar curvature. Hence it appears to be applicable to Ricciflat spacetimes and emerges as
a valuable tool for setting up flat holography. Such a smooth behaviour is not generic, as
in most coordinate systems switching off the scalar curvature for an Einstein space leads
to plain Minkowski spacetime.1
The observations above suggest that it is relevant to wonder whether a Ricciflat
spacetime admits a dual fluid description. This can be recast into two sharp questions:
1. Which surface S would replace the AdS conformal boundary I , and what is the
geometry that this new boundary should be equipped with?
2. Which are the degrees of freedom hosted by S and succeeding the relativisticfluid
energymomentum tensor, and what is the dynamics these degrees of freedom obey?
Many proposals have been made for answering these questions. Most of them were
inspired by the seminal work [17, 18], where NavierStokes equations were shown to
capture the dynamics of blackhole horizon perturbations. This result is taken as the crucial
evidence regarding the deep relation between gravity, without cosmological constant, and
fluid dynamics.
A more recent approach has associated Ricciflat spacetimes in d + 1 dimensions with
ddimensional fluids [19–24]. This is based on the observation that the BrownYork
energymomentum tensor on a Rindler hypersurface of a flat metric has the form of a perfect
fluid [25]. In this particular framework, one can consider a nonrelativistic limit, thus
showing that the NavierStokes equations coincide with Einstein’s equations on the Rindler
hypersurface. Paradoxically, it has simultaneously been argued that all information can be
stored in a relativistic ddimensional fluid.
Outside the realm of fluid interpretation, and on the more mathematical side of the
problem, some solid works regarding flat holography are [26–28] (see also [29]). The dual
1This phenomenon is well known in supergravity, when studying the gravity decoupling limit of scalar
manifolds. For this limit to be nontrivial, one has to chose an appropriate gauge (see [15, 16] for a recent
discussion and references).
– 2 –
theories reside at null infinity emphasizing the importance of the nulllike formalisms of [30–
32]. In this line of thought, results where also reached focusing on the expected symmetries,
in particular for the specific case of threedimensional bulk versus twodimensional
boundary [33–39].2 These achievements are not unconditionally transferable to four or higher
dimensions, and can possibly infer inaccurate expectations due to features holding
exclusively in three dimensions.
The above wanderings between relativistic and nonrelativistic fluid dynamics in
relation with Ricciflat spacetimes are partly due to the incomplete understanding on the rˆole
played by the null infinity. On the one hand, it has been recognized that the Ricciflat limit
is related to some contraction of the Poincar´e algebra [33–37, 40, 41]. On the other hand,
this observation was tempered by a potential confusion among the Carrollian algebra and
its dual contraction, the conformal Galilean algebra, as they both lead to the decoupling of
time. This phenomenon was exacerbated by the equivalence of these two algebras in two
dimensions, and has somehow obscured the expectations on the nature and the dynamics
of the relevant boundary degrees of freedom. Hence, although the idea of localizing the
latter on the spatial surface at null infinity was suggested (as e.g. in [42–45]), their
description has often been accustomed to the relativisticfluid or the conformalfieldtheory
approaches, based on the revered energymomentum tensor and its conservation law.3
From this short discussion, it is clear that the attempts implemented so far follow
different directions without clear overlap and common views. Although implicitly addressed
in the literature, the above two questions have not been convincingly answered, and the
treatment of boundary theories in the zero cosmological constant limit remains nowadays
tangled.
In this work we make a precise statement, which clarifies unquestionably the situation.
Our starting point is a fourdimensional bulk Einstein spacetime with Λ = −3k2, dual to
a boundary relativistic fluid. In this setup, we consider the k → 0 limit, which has the
following features:
• The derivative expansion is generically well behaved. We will call its limit the flat
derivative expansion. Under specified conditions it can be resummed in a closed form.
• Inside the boundary metric, and in the complete boundary fluid dynamics, k plays
the rˆole of velocity of light. Its vanishing is thus a Carrollian limit.
• The boundary is the twodimensional spatial surface S emerging as the future null
infinity of the limiting Ricciflat bulk spacetime. It replaces the AdS conformal
boundary and is endowed with a Carrollian geometry i.e. is covariant under Carrollian
diffeomorphisms.
• The degrees of freedom hosted by this surface are captured by a conformal Carrollian
fluid : energy density and pressure related by a conformal equation of state, heat
2Reference [37] is the first where a consistent and nontrivial k → 0 limit was taken, mapping the
3This is manifest in the very recent work of ref. [46].
currents and traceless viscous stress tensors. These macroscopic degrees of freedom
obey conformal Carrollian fluid dynamics.
Any twodimensional conformal Carrollian fluid hosted by an arbitrary spatial surface S ,
and obeying conformal Carrollian fluid dynamics on this surface, is therefore mapped onto
a Ricciflat fourdimensional spacetime using the flat derivative expansion. The latter is
invariant under boundary Weyl transformations. Under a set of resummability conditions
involving the Carrollian fluid and its host S , this derivative expansion allows to reconstruct
exactly algebraically special Ricciflat spacetimes. The results summarized above answer
in the most accurate manner the two questions listed earlier.
Carrollian symmetry has sporadically attracted attention following the pioneering work
or ref. [47], where the Carroll group emerged as a new contraction of the Poincar´e group: the
ultrarelativistic contraction, dual to the usual nonrelativistic one leading to the Galilean
group. Its conformal extensions were explored latterly [48–51], showing in particular its
relationship to the BMS group, which encodes the asymptotic symmetries of asymptotically
flat spacetimes along a null direction [53–56].4
It is therefore quite natural to investigate on possible relationships between Carrollian
asymptotic structure and flat holography and, by the logic of fluid/gravity correspondence,
to foresee the emergence of Carrollian hydrodynamics rather than any other, relativistic
or Galilean fluid. Nonetheless searches so far have been oriented towards the nearhorizon
membrane paradigm, trying to comply with the inevitable BMS symmetries as in [59, 60].
The power of the derivative expansion and its flexibility to handle the zerok limit has
been somehow dismissed. This expansion stands precisely at the heart of our method. Its
actual implementation requires a comprehensive approach to Carrollian hydrodynamics,
as it emanates from the ultrarelativistic limit of relativistic fluid dynamics, made recently
available in [52].
The aim of the present work is to provide a detailed analysis of the various
statements presented above, and exhibit a precise expression for the Ricciflat line element as
reconstructed from the boundary Carrollian geometry and Carrollian fluid dynamics. As
already stated, the tool for understanding and implementing operationally these ideas is
the derivative expansion and, under conditions, its resummed version. For this reason,
section 2 is devoted to its thorough description in the framework of ordinary antide Sitter
fluid/gravity holography. This section includes the conditions, stated in a novel fashion
with respect to [12, 13], for the expansion to be resummed in a closed form, representing
generally an Einstein spacetime of algebraically special Petrov type.
In section 3 we discuss how the Carrollian geometry emerges at null infinity and
describe in detail conformal Carrollian hydrodynamics following [52]. The formulation of the
Ricciflat derivative expansion is undertaken in section 4. Here we discuss the important
issue of resumming in a closed form the generic expansion. This requires the investigation
of another uncharted territory: the higherderivative curvaturelike Carrollian tensors. The
Carrollian geometry on the spatial boundary S is naturally equipped with a (conformal)
Carrollian connection, which comes with various curvature tensors presented in section 3.
4Carroll symmetry has also been explored in connection to the tensionlessstring limit, see e.g. [57, 58].
– 4 –
The relevant object for discussing the resummability in the antide Sitter case is the Cotton
tensor, as reviewed in section 2. It turns out that this tensor has welldefined Carrollian
descendants, which we determine and exploit. With those, the resummability conditions
are wellposed and set the framework for obtaining exact Ricciflat spacetimes in a closed
form from conformalCarrollianfluid data.
In order to illustrate our results, we provide examples starting from section 3 and
pursuing systematically in section 5. Generic Carrollian perfect fluids are meticulously
studied and shown to be dual to the general Ricciflat KerrTaubNUT family. The non
perfect Carrollian fluid called RobinsonTrautman fluid is discussed both as the limiting
RobinsonTrautman relativistic fluid (section 3), and alternatively from Carrollian first
This is a geometric quantity, which, if absent, makes possible for using holomorphic
coordinates. In appendix A, we gather the relevant formulas in this class of coordinates.
2
Fluid/gravity in asymptotically locally AdS spacetimes
We present here an executive summary of the holographic reconstruction of
fourdimensional asymptotically locally antide Sitter spacetimes from threedimensional relativistic
boundary fluid dynamics. The tool we use is the fluidvelocity derivative expansion. We
show that exact Einstein spacetimes written in a closed form can arise by resumming this
expansion. It appears that the key conditions allowing for such an explicit resummation
are the absence of shear in the fluid flow, as well as the relationship among the nonperfect
components of the fluid energymomentum tensor (i.e. the heat current and the viscous
stress tensor) and the boundary Cotton tensor.
2.1
The derivative expansion
The spirit.
Due to the FeffermanGraham ambient metric construction [61],
asymptotically locally antide Sitter fourdimensional spacetimes are determined by a set of
indetensor T = Tμν dxμdxν , symmetric (Tμν = Tνμ), traceless (T μμ = 0) and conserved:
pendent boundary data, namely a threedimensional metric ds2 = gμν dxμdxν and a rank2
∇μTμν = 0.
(2.1)
Perhaps the most well known subclass of asymptotically locally AdS spacetimes are
those whose boundary metrics are conformally flat (see e.g. [62, 63]). These are
asymptotically globally antide Sitter. The asymptotic symmetries of such spacetimes comprise the
finite dimensional conformal group, i.e. SO(3, 2) in four dimensions [64], and AdS/CFT
is at work giving rise to a boundary conformal field theory. Then, the rank2 tensor Tμν
is interpreted as the expectation value over a boundary quantum state of the
conformalfieldtheory energymomentum tensor. Whenever hydrodynamic regime is applicable, this
– 5 –
We assume local thermodynamic equilibrium with p the local pressure and ε the local
energy density:
ε =
1
k2 Tμν uμuν .
A localequilibrium thermodynamic equation of state p = p(T ) is also needed for completing
the system, and we omit the chemical potential as no independent conserved current, i.e.
no gauge field in the bulk, is considered here.
The symmetric viscous stress tensor τμν and the heat current qμ are purely transverse:
μ
u τμν = 0,
μ
u qμ = 0,
qν = −εuν − uμTμν .
approach gives rise to the socalled fluid/gravity correspondence and all its important
spinoffs (see [4] for a review).
For a long time, all the work on fluid/gravity correspondence was confined to
asymptotically globally AdS spacetimes, hence to holographic boundary fluids that flow on
conformally flat backgrounds. In a series of works [
9–14
] we have extended the fluid/gravity
correspondence into the realm of asymptotically locally AdS4 spacetimes. In the following,
we present and summarize our salient findings.
The energymomentum tensor.
Given the energymomentum tensor of the boundary
fluid and assuming that it represents a state in a hydrodynamic regime, one should be able
to pick a boundary congruence u, playing the rˆole of fluid velocity. Normalizing the latter
as5 kuk2 = −k2 we can in general decompose the energymomentum tensor as
Tμν = (ε + p) uμuν + pgμν + τμν +
k2
uμqν +
k2
uν qμ
derivatives, the coefficients of which characterize the transport phenomena occurring in the
fluid. In firstorder hydrodynamics
where hμν is the projector onto the space transverse to the velocity field:
5This unconventional normalization ensures that the derivative expansion is wellbehaved in the k → 0
are the acceleration (transverse), the expansion, the shear and the vorticity (both ranktwo
tensors are transverse and traceless). As usual, η, ζ are the shear and bulk viscosities, and
HJEP07(218)65
κ is the thermal conductivity.
It is customary to introduce the vorticity twoform
ω =
1
as well as the HodgePoincar´e dual of this form, which is proportional to u (we are in 2 + 1
kγu = ⋆ω ⇔ kγuμ =
1
2 ημνσωνσ,
−gǫμνσ. In this expression γ is a scalar, that can also be expressed as
dimensions):
where ημνσ = √
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
(2.18)
ν
aμ = u ∇ν uμ,
σμν = ∇(μuν) +
ωμν = ∇[μuν] +
,
– 7 –
In three spacetime dimensions and in the presence of a vector field, one naturally
defines a fully antisymmetric twoindex tensor as
obeying
With this tensor the vorticity reads:
Weyl covariance, Weyl connection and the Cotton tensor. In the case when
the boundary metric gμν is conformally flat, it was shown that using the above set of
boundary data it is possible to reconstruct the fourdimensional bulk Einstein spacetime
order by order in derivatives of the velocity field [3–6]. The guideline for the spacetime
reconstruction based on the derivative expansion is Weyl covariance: the bulk geometry
should be insensitive to a conformal rescaling of the boundary metric (weight −2)
6Our conventions for (anti)symmetrization are: A(μν) = 12 (Aμν + Aνμ) and A[μν] = 21 (Aμν − Aνμ).
which should correspond to a bulk diffeomorphism and be reabsorbed into a redefinition of
the radial coordinate: r → B r. At the same time, uμ is traded for uBμ (velocity oneform),
ωμν for ωBμν (vorticity twoform) and Tμν for BTμν . As a consequence, the pressure and
energy density have weight 3, the heatcurrent qμ weight 2, and the viscous stress tensor
Covariantization with respect to rescaling requires to introduce a Weyl connection
A =
1
k2
Θ
a − 2
u ,
which transforms as A → A − d ln B. Ordinary covariant derivatives ∇ are thus traded
for Weyl covariant ones D = ∇ + w A, w being the conformal weight of the tensor under
consideration. We provide for concreteness the Weyl covariant derivative of a weightw
Dν vμ = ∇ν vμ + (w + 1)Aν vμ + Aμvν − gμν Aρvρ.
The Weyl covariant derivative is metric with effective torsion:
τμν weight 1.
oneform:7
form vμ:
where
is Weylinvariant.
Weyl covariant Riemann tensor
Dρgμν = 0,
(DμDν − Dν Dμ) f = wf Fμν ,
Fμν = ∂μAν − ∂ν Aμ
Commuting the Weylcovariant derivatives acting on vectors, as usual one defines the
(DμDν − Dν Dμ) V ρ = Rρσμν V σ + wV ρFμν
(V ρ are weightw) and the usual subsequent quantities. In three spacetime dimensions, the
covariant Ricci (weight 0) and the scalar (weight 2) curvatures read:
Rμν = Rμν + ∇ν Aμ + AμAν + gμν ∇λA
λ
− AλA
λ
− Fμν ,
R = R + 4∇μA
μ
− 2AμA .
μ
The Weylinvariant Schouten tensor8 is
1
Sμν = Rμν − 4
Rgμν = Sμν + ∇ν Aμ + AμAν − 2
1
AλAλgμν − Fμν .
7The explicit form of A is obtained by demanding Dμuμ = 0 and uλDλuμ = 0.
8The ordinary Schouten tensor in three spacetime dimensions is given by Rμν − 14 Rgμν.
– 8 –
(2.19)
(2.20)
(2.21)
(2.22)
(2.23)
(2.24)
(2.25)
(2.26)
(2.27)
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Other Weylcovariant velocityrelated quantities are
Dμuν = ∇μuν +
1
k2 uμaν − 2
hμν
Θ
= σμν + ωμν ,
Dν ωνμ = ∇ν ωνμ,
Dν ηνμ = 2γuμ,
uλRλμ = D
λ σλμ − ωλμ − uλFλμ,
(2.28)
(2.29)
(2.30)
(2.31)
(2.32)
(2.33)
(2.34)
(2.35)
(2.36)
(2.37)
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of weights −1, 1, 0 and 1 (the scalar vorticity γ has weight 1).
The remarkable addition to the fluid/gravity dictionary came with the realization that
the derivative expansion can be used to reconstruct Einstein metrics which are
asymptotically locally AdS. For the latter, the boundary metric has a non zero Cotton tensor [
9–13
].
The Cotton tensor is generically a threeindex tensor with mixed symmetries. In three
dimensions, which is the case for our boundary geometry, the Cotton tensor can be dualized
into a twoindex, symmetric and traceless tensor. It is defined as
Cμν = ημρσDρ (Sνσ + Fνσ) = ημρσ
∇ρ
R
Rνσ − 4 gνσ
.
The Cotton tensor is Weylcovariant of weight 1 (i.e. transforms as Cμν → B Cμν ), and is
identically conserved:
DρCρν = ∇ρCρν = 0,
sharing thereby all properties of the energymomentum tensor. Following (2.2) we can
decompose the Cotton tensor into longitudinal, transverse and mixed components with
respect to the fluid velocity u:9
Cμν =
2
3c uμuν +
k
ck
2
gμν −
k
cμν +
uμcν +
k
uν cμ
k
.
Such a decomposition naturally defines the weight3 Cotton scalar density
c =
1
k3 Cμν uμuν ,
as the longitudinal component. The symmetric and traceless Cotton stress tensor cμν and
the Cotton current cμ (weights 1 and 2, respectively) are purely transverse:
and obey
cμμ = 0,
μ
u cμν = 0,
μ
u cμ = 0,
cμν = −khρμhσν Cρσ +
ck2
2
hμν ,
cν = −cuν −
uμCμν .
k
9Notice that the energymomentum tensor has an extra factor of k with respect to the Cotton tensor,
see (2.60), due to their different dimensions.
– 9 –
One can use the definition (2.32) to further express the Cotton density, current and
stress tensor as ordinary or Weyl derivatives of the curvature. We find
c =
k2
1 uν ησρDρ (Sνσ + Fνσ) ,
cν = ηρσDρ (Sνσ + Fνσ) − cuν ,
cμν = −hλμ (kηνρσ
− uν ηρσ) Dρ (Sλσ + Fλσ) +
ck2
2
hμν .
The bulk Einstein derivative expansion.
Given the ingredients above, the leading
terms in a 1r expansion for a fourdimensional Einstein metric are of the form:10
HJEP07(218)65
u
u
2
ds2bulk = 2 k2 (dr + rA) + r2ds2 +
S
k4
1
1 − 2k4r2 ωαβωαβ
8πGTλμuλuμ
k2
r +
Cλμuλημνσωνσ
2k4
+ terms with σ, σ2, ∇σ, . . . + O D 4u .
In this expression
• S is a Weylinvariant tensor:
S = Sμν dxμdxν = −2uDν ωνμdxμ − ωμλωλν dxμdxν − u
2
R
2
;
• the boundary metric is parametrized `a la RandersPapapetrou:
ds2 = −k
2 Ωdt − bidxi 2
+ aij dxidxj ;
u =
1
Ω ∂t ⇔ u = −k
2 Ωdt − bidxi ,
• the boundary conformal fluid velocity field and the corresponding one form are
i.e. the fluid is at rest in the frame associated with the coordinates in (2.43) — this
is not a limitation, as one can always choose a local frame where the fluid is at rest,
in which the metric reads (2.43) (with Ω, bi and aij functions of all coordinates);
• ωμν is the vorticity of u as given in (2.11), which reads:
ω =
1
k
2
2
∂ibj +
1
Ω bi∂j Ω +
1
Ω bi∂tbj dxi ∧ dxj ;
(2.45)
• γ2 = 12 aikajl ∂[ibj] + Ω1 b[i∂j]Ω + Ω1 b[i∂tbj]
∂[kbl] + Ω1 b[k∂l]Ω + Ω1 b[k∂tbl] ;
10We have traded here the usual advancedtime coordinate used in the quoted literature on fluid/gravity
correspondence for the retarded time, spelled t (see (2.44)).
(2.38)
(2.39)
(2.40)
(2.41)
(2.42)
(2.43)
(2.44)
• the expansion and acceleration are
leading to the Weyl connection
with a the determinant of aij ;
all vanish due to (2.4);
Θ =
• k12 Tμν uμuν is the energy density ε of the fluid (see (2.3)), and in the
Randers
Papapetrou frame associated with (2.43), (2.44), q0, τ00, τ0i = τi0 entering in (2.2)
• 2k14 Cλμuλημνσωνσ = cγ, where we have used (2.13) and (2.35), and similarly c0 =
c00 = c0i = ci0 = 0 as a consequence of (2.36) with (2.43), (2.44);
• σ, σ2, ∇σ stand for the shear of u and combinations of it, as computed from (2.10):
σ =
1
2Ω
We have not exhibited explicitly shearrelated terms because we will ultimately assume the
absence of shear for our congruence. This raises the important issue of choosing the fluid
velocity field, not necessary in the FeffermanGraham expansion, but fundamental here. In
relativistic fluids, the absence of sharp distinction between heat and matter fluxes leaves a
freedom in setting the velocity field. This choice of hydrodynamic frame is not completely
arbitrary though, and one should stress some reservations, which are often dismissed, in
particular in the already quoted fluid/gravity literature.
As was originally exposed in [65] and extensively discussed e.g. in [7], the fluidvelocity
ambiguity is well posed in the presence of a conserved current J, naturally decomposed
into a longitudinal perfect piece and a transverse part:
J
μ = ̺uμ + jμ.
(2.50)
The velocity freedom originates from the redundancy in the heat current q and the
nonperfect piece of the matter current j. One may therefore set j = 0 and reach the Eckart
frame. Alternatively q = 0 defines the LandauLifshitz frame. In the absence of matter
current, nothing guarantees that one can still move to the LandauLifshitz frame, and
setting q = 0 appears as a constraint on the fluid, rather than a choice of frame for
describing arbitrary fluids. This important issue was recently discussed in the framework
of holography [66], from which it is clear that setting q = 0 in the absence of a conserved
current would simply inhibit certain classes of Einstein spaces to emerge holographically
from boundary data, and possibly blur the physical phenomena occurring in the fluids
under consideration. Consequently, we will not make any such assumption, keeping the
heat current as part of the physical data.
We would like to close this section with an important comment on asymptotics. The
reconstructed bulk spacetime can be asymptotically locally or globally antide Sitter. This
property is read off directly inside terms appearing at designated orders in the radial
expansion, and built over specific boundary tensors. For d + 1dimensional boundaries,
the boundary energymomentum contribution first appears at order rd1−1 , whereas the
boundary Cotton tensor11 emerges at order r12 . This behaviour is rooted in the
EddingtonFinkelstein gauge used in (2.41), but appears also in the slightly different Bondi gauge.
It is however absent in the FeffermanGraham coordinates, where the Cotton cannot be
possibly isolated in the expansion.
Resummation and exact Einstein spacetimes in closed form. In order to further
probe the derivative expansion (2.41), we will impose the fluid velocity congruence be
shearless. This choice has the virtue of reducing considerably the number of terms compatible
with conformal invariance in (2.41), and potentially making this expansion resummable,
thus leading to an Einstein metric written in a closed form. Nevertheless, this shearless
condition reduces the class of Einstein spacetimes that can be reconstructed
holographically to the algebraically special ones [
10–14
]. Going beyond this class is an open problem
that we will not address here.
Following [
6, 10–14
], it is tempting to try a resummation of (2.41) using the following
substitution:
with
γ
2
1 − r2 → ρ2
r
2
ρ2 = r2 + γ2.
The resummed expansion would then read
u
dsr2es. Einstein = 2 k2 (dr + rA) + r2ds2 +
S
k4 +
u
2
k4ρ2 (8πGεr + cγ) ,
which is indeed written in a closed form. Under the conditions listed below, the
metric (2.53) defines the line element of an exact Einstein space with Λ = −3k2.
• The congruence u is shearless. This requires (see (2.49))
∂taij = aij ∂t ln √a.
Actually (2.54) is equivalent to ask that the twodimensional spatial section S defined
at every time t and equipped with the metric dℓ2 = aij dxidxj is conformally flat.
This may come as a surprise because every twodimensional metric is conformally
11Actually, the object appearing in generic dimension is the Weyl divergence of the boundary Weyl tensor,
which contains also the Cotton tensor (see [67] for a preliminary discussion on this point).
(2.51)
(2.52)
(2.53)
(2.54)
flat. However, aij generally depends on space x and time t, and the transformation
required to bring it in a form proportional to the flatspace metric might depend on
time. This would spoil the threedimensional structure (2.43) and alter the a priori
given u. Hence, dℓ2 is conformally flat within the threedimensional spacetime (2.43)
under the condition that the transformation used to reach the explicit conformally
flat form be of the type x′ = x′(x). This exists if and only if (2.54) is satisfied.12
Under this condition, one can always choose ζ = ζ(x), ζ¯ = ζ¯(x) such that
dℓ2 = aij dxidxj =
2
with P = P (t, ζ, ζ¯) a real function. Even though this does not hold for arbitrary
u = ∂Ωt , one can show that there exists always a congruence for which it does [68],
and this will be chosen for the rest of the paper.
• The heat current of the boundary fluid introduced in (2.2) and (2.4) is identified with
the transversedual of the Cotton current defined in (2.34) and (2.37). The Cotton
current being transverse to u, it defines a field on the conformally flat twosurface S ,
the existence of which is guaranteed by the absence of shear. This surface is endowed
with a natural hodge duality mapping a vector onto another, which can in turn be
lifted back to the threedimensional spacetime as a new transverse vector. This whole
process is taken care of by the action of ηνμ defined in (2.15):
qμ =
1
8πG
1
8πG
ηνμcν =
ηνμηρσDρ (Sνσ + Fνσ) ,
¯
coordinates ζ, ζ¯ as in (2.55)13 leads to ηζζ = i and ηζζ¯ = −i, and thus
where we used (2.39) in the last expression. Using holomorphic and antiholomorphic
q =
i
8πG
cζ dζ − cζ¯dζ¯ .
• The viscous stress tensor of the boundary conformal fluid introduced in (2.2) is
identified with the transversedual of the Cotton stress tensor defined in (2.34) and (2.37).
Following the same pattern as for the heat current, we obtain:
1
τμν = − 8πGk2 ηρμcρν
=
1
8πGk2
− 21 uλημν ηρσ + ηλμ (kηνρσ
− uν ηρσ) Dρ (Sλσ + Fλσ) ,
where we also used (2.40) in the last equality. The viscous stress tensor τμν is
transverse symmetric and traceless because these are the properties of the Cotton stress
tensor cμν . Similarly, we find in complex coordinates:
i
τ = − 8πGk2 cζζ dζ2 − cζ¯ζ¯dζ¯2 .
12A peculiar subclass where this works is when ∂t is a Killing field.
13Orientation is chosen such that in the coordinate frame η0ζζ¯ = √−gǫ0ζζ¯ = PiΩ2 , where x0 = kt.
(2.56)
(2.57)
(2.58)
(2.59)
• The energymomentum tensor defined in (2.2) with p = 2ε , heat current as in (2.56)
and viscous stress tensor as in (2.58) must be conserved, i.e. obey eq. (2.1). These
are differential constraints that from a bulk perspective can be thought of as a
generalization of the Gauss law.
Identifying parts of the energymomentum tensor with the Cotton tensor may be
viewed as setting integrability conditions, similar to the electricmagnetic duality conditions
in electromagnetism, or in Euclidean gravitational dynamics. As opposed to the latter, it
is here implemented in a rather unconventional manner, on the conformal boundary.
It is important to emphasize that the conservation equations (2.1) concern all
boundary data. On the fluid side the only remaining unknown piece is the energy density ε(x),
whereas for the boundary metric Ω(x), bi(x) and aij (x) are available and must obey (2.1),
together with ε(x). Given these ingredients, (2.1) turns out to be precisely the set of
equations obtained by demanding bulk Einstein equations be satisfied with the metric (2.53).
This observation is at the heart of our analysis.
The bulk algebraic structure and the physics of the boundary fluid. The pillars
of our approach are (i) the requirement of a shearless fluid congruence and (ii) the
identification of the nonperfect energymomentum tensor pieces with the corresponding Cotton
components by transverse dualization.
What does motivate these choices? The answer to this question is rooted to the Weyl
tensor and to the remarkable integrability properties its structure can provide to the system.
Let us firstly recall that from the bulk perspective, u is a manifestly null congruence
associated with the vector ∂r . One can show (see [13]) that this bulk congruence is also
geodesic and shearfree. Therefore, accordingly to the generalizations of the GoldbergSachs
theorem, if the bulk metric (2.41) is an Einstein space, then it is algebraically special, i.e.
of Petrov type II, III, D, N or O. Owing to the close relationship between the algebraic
structure and the integrability properties of Einstein equations, it is clear why the absence
of shear in the fluid congruence plays such an instrumental roˆle in making the tentatively
resummed expression (2.53) an exact Einstein space.
The structure of the bulk Weyl tensor makes it possible to go deeper in foreseeing how
the boundary data should be tuned in order for the resummation to be successful. Indeed
the Weyl tensor can be expanded for larger, and the dominant term ( r13 ) exhibits the
following combination of the boundary energymomentum and Cotton tensors [
69–73
]:
(2.60)
(2.61)
satisfying a conservation equation, analogue to (2.1)
For algebraically special spaces, these complexconjugate tensors simplify
considerably (see detailed discussions in [
10–14
]), and this suggests the transverse duality enforced
between the Cotton and the energymomentum nonperfect components. Using (2.57)
Tμ±ν = Tμν ± 8πGk
i
Cμν ,
∇μTμ±ν = 0.
and (2.59), we find indeed for the tensor T+ in complex coordinates:
T+ = ε +
ic
8πG
and similarly for T− obtained by complex conjugation with
.
ic
The bulk Weyl tensor and consequently the Petrov class of the bulk Einstein space are
encoded in the three complex functions of the boundary coordinates: ε+, cζ and cζζ.
The proposed resummation procedure, based on boundary relativistic fluid
dynamics of nonperfect fluids with heat current and stress tensor designed from the
boundary Cotton tensor, allows to reconstruct all algebraically special fourdimensional Einstein
spaces. The simplest correspond to a Cotton tensor of the perfect form [
10
]. The complete
class of Pleban´skiDemian´ski family [74] requires nontrivial bi with two commuting Killing
fields [13], while vanishing bi without isometry leads to the RobinsonTrautman Einstein
spaces [12]. For the latter, the heat current and the stress tensor obtained from the Cotton
by the transverse duality read:
1
1
q = − 16πG
∂ζKdζ + ∂ζ¯Kdζ¯ ,
τ =
8πGk2P 2 ∂ζ P 2∂t∂ζ ln P dζ2 + ∂ζ¯ P 2∂t∂ζ¯ ln P dζ¯2 ,
where K = 2P 2∂ζ¯∂ζ ln P is the Gaussian curvature of (2.55). With these data the
conservation of the energymomentum tensor (2.1) enforces the absence of spatial dependence in
ε = 2p, and leads to a single independent equation, the heat equation:
12M ∂t ln P + ΔK = 4∂tM.
This is the RobinsonTrautman equation, here expressed in terms of M (t) = 4πGε(t).
The boundary fluids emerging in the systems considered here have a specific physical
behaviour. This behaviour is inherited from the boundary geometry, since their excursion
away from perfection is encoded in the Cotton tensor via the transverse duality. In the
hydrodynamic frame at hand, this implies in particular that the derivative expansion of the
energymomentum tensor terminates at third order. Discussing this side of the holography
is not part of our agenda. We shall only stress that such an analysis does not require to
change hydrodynamic frame. Following [66], it is possible to show that the frame at hand is
the Eckart frame. Trying to discard the heat current in order to reach a
LandauLifshitzlike frame (as in [75–78] for RobinsonTrautman) is questionable, as already mentioned
earlier, because of the absence of conserved current, and distorts the physical phenomena
occurring in the holographic conformal fluid.
3
The Ricciflat limit I: Carrollian geometry and Carrollian fluids
The Ricciflat limit is achieved at vanishing k. Although no conformal boundary exists in
this case, a twodimensional spatial conformal structure emerges at null infinity. Since the
(2.62)
(2.63)
(2.64)
(2.65)
(2.66)
Einstein bulk spacetime derivative expansion is performed along null tubes, it provides the
appropriate arena for studying both the nature of the twodimensional “boundary” and the
dynamics of the degrees of freedom it hosts as “holographic duals” to the bulk Ricciflat
For vanishing k, time decouples in the boundary
geometry (2.43). There exist two decoupling limits, associated with two distinct contractions
of the Poincar´e group: the Galilean, reached at infinite velocity of light and referred to as
“nonrelativistic”, and the Carrollian, emerging at zero velocity of light [47] — often called
“ultrarelativistic”. In (2.43), k plays effectively the rˆole of velocity of light and k → 0 is
indeed a Carrollian limit.
This very elementary observation sets precisely and unambiguously the fate of
asymptotically flat holography: the reconstruction of fourdimensional Ricciflat spacetimes is
based on Carrollian boundary geometry.
The appearance of Carrollian symmetry, or better, conformal Carrollian symmetry at
null infinity of asymptotically flat spacetimes is not new [48–51]. It has attracted attention
in the framework of flat holography, mostly from the algebraic side [79, 80], or in relation
with its dual geometry emerging in the Galilean limit, known as NewtonCartan (see [81]).
The novelties we bring in the present work are twofold. On the one hand, the Carrollian
geometry emerging at null infinity is generally nonflat, i.e. it is not isometric under the
Carroll group, but under a more general group associated with a timedependent
positivedefinite spatial metric and a Carrollian time arrow, this general Carrollian geometry being
covariant under a subgroup of the diffeomorphisms dubbed Carrollian diffeomorphisms.
On the other hand, the Carrollian surface is the natural host for a Carrollian fluid,
zerok limit of the relativistic boundary fluid dual to the original Einstein space of which we
consider the flat limit. This Carrollian fluid must be considered as the holographic dual of
a Ricciflat spacetime, and its dynamics (studied in section 3.2) as the dual of gravitational
bulk dynamics at zero cosmological constant. From the hydrodynamical viewpoint, this
gives a radically new perspective on the subject of flat holography.
The Carrollian geometry: connection and curvature. The Carrollian geometry
consists of a spatial surface S endowed with a positivedefinite metric
dℓ2 = aij dxidxj ,
(3.1)
and a Carrollian time t ∈ R.14 The metric on S is generically timedependent: aij =
aij (t, x). Much like a Galilean space is observed from a spatial frame moving with
respect to a local inertial frame with velocity w, a Carrollian frame is described by a form
×
14We are genuinely describing a spacetime R
S endowed with a Carrollian structure, and this is actually
how the boundary geometry should be spelled. In order to make the distinction with the relativistic
pseudoRiemannian threedimensional spacetime boundary I of AdS bulks, we quote only the spatial surface
S when referring to the Carrollian boundary geometry of a Ricciflat bulk spacetime. For a complete
description of such geometries we recommend [82].
b = bi(t, x) dxi. The latter is not a velocity because in Carrollian spacetimes motion is
forbidden. It is rather an inverse velocity, describing a “temporal frame” and plays a dual
rˆole. A scalar Ω(t, x) is also introduced (as in the Galilean case, see [52] — this reference
will be useful along the present section), as it may naturally arise from the k → 0 limit.
We define the Carrollian diffeomorphisms as
t′ = t′(t, x)
and
′
x = x (x)
′
with Jacobian functions
′
∂t
∂t
,
J (t, x) =
ji(t, x) =
′
∂t
∂xi
,
Jji(x) =
∂xi′
∂xj
.
Those are the diffeomorphisms adapted to the Carrollian geometry since under such
transformations, dℓ2 remains a positivedefinite metric (it does not produce terms involving dt′).
Indeed,
a′ij = aklJ −1ikJ −1l,
j
b′k =
bi +
Ω
J ji J −1ki ,
′
Ω =
Ω
J
,
whereas the time and space derivatives become
∂t =
1
J ∂t,
∂j′ = J −1i ∂i − J ∂t .
i
j
j
We will show in a short while that the Carrollian fluid equations are precisely covariant
under this particular set of diffeomorphisms.
Expression (3.5) shows that the ordinary exterior derivative of a scalar function does
not transform as a form. To overcome this issue, it is desirable to introduce a Carrollian
derivative as
transforming as
∂ˆi = ∂i + Ωi ∂t,
b
∂ˆi = J −1ij ∂ˆj .
′
Acting on scalars this provides a form, whereas for any other tensor it must be covariantized
by introducing a new connection for Carrollian geometry, called LeviCivitaCarroll
connection, whose coefficients are the ChristoffelCarroll symbols,15
γˆjik =
ail
2
∂ˆj alk + ∂ˆkalj − ∂ˆlajk
= γjik + cijk.
they are identical to ∂ˆi:
The LeviCivitaCarroll covariant derivative acts symbolically as ∇ˆ = ∂ˆ + γˆ. It is
metric and torsionless: ∇ˆ iajk = 0, tˆkij = 2γˆ[kij] = 0. There is however an effective torsion,
since the derivatives ∇ˆ i do not commute, even when acting of scalar functions Φ — where
[∇ˆ i, ∇ˆj ]Φ =
̟ij ∂tΦ.
2
Ω
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
acceleration oneform ϕi:
Since the original relativistic fluid is at rest, the kinematical “inversevelocity” variable
potentially present in the Carrollian limit vanishes.16 Hence the various kinematical
quantities such as the vorticity and the acceleration are purely geometric and originate from the
temporal Carrollian frame used to describe the surface S . As we will see later, they turn
out to be k → 0 counterparts of their relativistic homologues defined in (2.9), (2.10), (2.11)
(see also (3.14) for the expansion and shear).
The time derivative transforms as in (3.5), and acting on any tensor under Carrollian
diffeomorphisms, it provides another tensor. This ordinary time derivative has nonetheless
an unsatisfactory feature: its action on the metric does not vanish. One is tempted therefore
to set a new time derivative ∂ˆt such that ∂ˆtajk = 0, while keeping the transformation rule
under Carrollian diffeomorphisms: ∂ˆt′ = J1 ∂ˆt. This is achieved by introducing a “temporal
Carrollian connection”
which allows us to define the time covariant derivative on a vector field:
γˆij =
2Ω
1 aik∂takj ,
Ω1 ∂ˆtV i =
1
Ω ∂tV i + γˆij V j ,
while on a scalar the action is as the ordinary time derivative: ∂ˆtΦ = ∂tΦ. Leibniz rule
allows extending the action of this derivative to any tensor.
Calling γˆij a connection is actually misleading because it transforms as a genuine
tensor under Carrollian diffeomorphisms: γˆ′kj = JnkJ −1jmγˆnm. Its trace and traceless parts
have a welldefined kinematical interpretation, as the expansion and shear, completing the
acceleration and vorticity introduced earlier in (3.10), (3.11):
θ = γˆii =
ξij = γˆij − 21 δji θ =
2Ω
1 aik ∂takj − akj ∂t ln √
a .
We can define the curvature associated with a connection, by computing the
commutator of covariant derivatives acting on a vector field. We find
where
h∇ˆ k, ∇ˆ li V i = rˆijklV j + ̟kl Ω ∂tV i,
2
rˆ jkl = ∂ˆkγˆlj − ∂ˆlγˆkij + γˆkimγˆlmj − γˆlimγˆkmj
i i
is a genuine tensor under Carrollian diffeomorphisms, the RiemannCarroll tensor.
16A Carrollian fluid is always at rest, but could generally be obtained from a relativistic fluid moving
at vi = k2βi + O k4 . In this case, the “inverse velocity” β
i would contribute to the kinematics and the
dynamics of the fluid (see [52]). Here, vi = 0 before the limit k → 0 is taken, so βi = 0.
• the energy density ε(t, x) and the pressure p(t, x), related here through a conformal
equation of state ε = 2p;
• the heat currents Q = Qi(t, x)dxi and π = πi(t, x)dxi;
• the viscous stress tensors Σ = Σij (t, x) dxidxj and Ξ = Ξij (t, x)dxidxj .
The latter quantities are inherited from the relativistic ones (see (2.2)) as the following
limits:
and p.
Qi = kli→m0 qi,
Σij = − kli→m0 k2τij ,
1
πi = lim
k→0 k2 (qi − Qi) ,
Ξij = − kli→m0
τij +
1
k2 Σij .
Compared with the corresponding ones in the Galilean fluids, they are doubled because
two orders seem to be required for describing the Carrollian dynamics. They obey
Σij = Σji,
Σii = 0,
Ξij = Ξji,
Ξii = 0.
The Carrollian energy and pressure are just the zerok limits of the corresponding
relativistic quantities. In order to avoid symbols inflation, we have kept the same notation, ε
All these objects are Weylcovariant with conformal weights 3 for the pressure and
energy density, 2 for the heat currents, and 1 for the viscous stress tensors (when all indices
are lowered). They are welldefined in all examples we know from holography. Ultimately
they should be justified within a microscopic quantum/statistical approach, missing at
present since the microscopic nature of a Carrollian fluid has not been investigated so far,
except for [52], where some elementary issues were addressed.
Following this reference, the equations for a Carrollian fluid are as follows:
• a set of two scalar equations, both weight4 Weylcovariant:
− Ω
1 Dˆtε − DˆiQi + Ξij ξij = 0,
Σij ξij = 0;
• two vector equations, Weylcovariant of weight 3:
Dˆj p + 2Qi̟ij +
Ω
Ω
1 Dˆtπj − DˆiΞij + πiξij = 0,
1 DˆtQj − DˆiΣij + Qiξij = 0.
Equation (3.49) is the energy conservation, whereas (3.50) sets a geometrical constraint
on the Carrollian viscous stress tensor Σij . Equations (3.51) and (3.52) are dynamical
equations involving the pressure p = 2ε
, the heat currents Qi and πi, and the viscous
stress tensors Σij and Ξij . They are reminiscent of a momentum conservation, although
somewhat degenerate due to the absence of fluid velocity.
(3.46)
(3.47)
(3.48)
(3.49)
(3.50)
(3.51)
(3.52)
An example of Carrollian fluid. The simplest nontrivial example of a Carrollian fluid
is obtained as the Carrollian limit of the relativistic RobinsonTrautman fluid, studied at
the end of section 2.2 (see also [66] and [52] for the relativistic and Carrollian approaches,
respectively).
The geometric Carrollian data are in this case
dℓ2 =
2
bi = 0 and Ω = 1. Hence the Carrollian shear vanishes (ξij = 0), whereas the expansion
Similarly ̟ij = 0, ϕi = 0, ϕij = 0, and using results from appendix A, we find
(in fact Kˆ = Kˆ = K), while
Kˆ = 2P 2∂ζ¯∂ζ ln P,
Aˆ = 0
Rˆζ¯ = ∂ζ¯∂t ln P,
Rˆζ¯ = ∂ζ¯∂t ln P.
From the relativistic heat current q and viscous stress tensor τ displayed in (2.64)
and (2.65), we obtain the Carrollian descendants:21
1
1
Q = − 16πG
∂ζ Kdζ + ∂ζ¯Kdζ¯ ,
Σ = − 8πGP 2 ∂ζ P 2∂t∂ζ ln P dζ2 + ∂ζ¯ P 2∂t∂ζ¯ ln P dζ¯2 ,
π = 0,
Ξ = 0.
Due to the absence of shear, the hydrodynamic equation (3.50) is identically satisfied,
whereas (3.49), (3.51), (3.52) are recast as:
3ε∂t ln P − ∂tε − ∇iQi = 0,
∂tQi − 2Qi∂t ln P − ∇j Σji = 0.
In agreement with the relativistic RobinsonTrautman fluid, the pressure p (and so the
energy density, since the fluid is conformal) must be spaceindependent. Furthermore, as
expected from the relativistic case, eq. (3.61) is satisfied with Qi and Σij given in (3.57)
and (3.58). Hence we are left with a single nontrivial equation, eq. (3.59), the heat equation
of the Carrollian fluid:
3ε∂t ln P − ∂tε +
ΔK = 0
1
16πG
with Δ = ∇j ∇j the Laplacian operator on S .
Equation (3.62) is exactly RobinsonTrautman’s, eq. (2.66). We note that the
relativistic and the Carrolian dynamics lead to the same equations — and hence to the same
21Notice a useful identity: ∂t
2
∂ζ P
P
= P12 ∂ζ P 2∂t∂ζ ln P .
(3.53)
(3.54)
(3.55)
(3.56)
(3.57)
(3.58)
(3.59)
(3.60)
(3.61)
(3.62)
solutions ε = ε(t). This is specific to the case under consideration, and it is actually
expected since the bulk Einstein equations for a geometry with a shearless and vorticityfree
null congruence lead to the RobinsonTrautman equation, irrespective of the presence of
a cosmological constant, Λ = −3k2: asymptotically locally AdS or locally flat spacetimes
lead to the same dynamics. This is not the case in general though, because there is no
reason for the relativistic dynamics to be identical to the Carrollian (see [52] for a detailed
account of this statement). For example, when switching on more data, as in the case of
the Pleban´skiDemian´ski family, where all bi, ϕi, ̟ij , as well as πi and Ξij , are on, the
Carrollian equations are different from the relativistic ones.
4
The Ricciflat limit II: derivative expansion and resummation
We can summarize our observations as follows. Any fourdimensional Ricciflat spacetime is
associated with a twodimensional spatial surface, emerging at null infinity and equipped
with a conformal Carrollian geometry. This geometry is the host of a Carrollian fluid,
obeying Carrollian hydrodynamics. Thanks to the relativisticfluid/AdSgravity duality,
one can also safely claim that, conversely, any Carrollian fluid evolving on a spatial
surface with Carrollian geometry is associated with a Ricciflat geometry. This conclusion is
reached by considering the simultaneous zerok limit of both sides of the quoted duality.
In order to make this statement operative, this limit must be performed inside the
derivative expansion. When the latter is resummable in the sense discussed in section 2.2, the
zerok limit will also affect the resummability conditions, and translate them in terms of
Carrollian fluid dynamics.
4.1
Back to the derivative expansion
Our starting point is the derivative expansion of an asymptotically locally AdS spacetime,
eq. (2.41). The fundamental question is whether the latter admits a smooth zerok limit.
We have implicitly assumed that the RandersPapapetrou data of the
threedimensional pseudoRiemannian conformal boundary I associated with the original Einstein
spacetime, aij , bi and Ω, remain unaltered at vanishing k, providing therefore directly the
Carrollian data for the new spatial twodimensional boundary S emerging at I +.22 Following
again the detailed analysis performed in [52], we can match the various threedimensional
Riemannian quantities with the corresponding twodimensional Carrollian ones:
u = −k2 (Ωdt − b)
(4.1)
22Indeed our ultimate goal is to set up a derivative expansion (in a closed resummed form under
appropriate assumptions) for building up fourdimensional Ricciflat spacetimes from a boundary Carrollian fluid,
irrespective of its AdS origin. For this it is enough to assume aij, bi and Ω kindependent (as in [52]), and
use these data as fundamental blocks for the Ricciflat reconstruction. It should be kept in mind, however,
that for general Einstein spacetimes, these may depend on k with welldefined limit and subleading terms.
Due to the absence of shear and to the particular structure of these solutions, the latter do not alter the
Carrollian equations. This occurs for instance in Pleban´skiDemian´ski or in the KerrTaubNUT subfamily,
which will be discussed in section 5.1. In the following, we avoid discussing this kind of subleading terms,
hence saving further technical developments.
and
̟ij dxi ∧ dxj ,
k
2
ω =
γ = ∗̟,
Θ = θ,
a = k2ϕidxi,
A = αidxi +
Ωdt,
σ = ξij dxidxj ,
θ
2
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
where the lefthandside quantities are Riemannian (given in eqs. (2.45), (2.46), (2.47),
(2.48), (2.49)), and the righthandside ones Carrollian (see (3.10), (3.11), (3.14), (3.20)).
In the list (4.2), we have dealt with the first derivatives, i.e. connexionrelated
quantities. We move now to secondderivative objects and collect the tensors relevant for the
derivative expansion, following the same pattern (Riemannian vs. Carrollian):
R =
1
k2 ξij ξij + 2Kˆ + 2k2 ∗ ̟2,
ωμλωλν dxμdxν = k4̟il̟lj dxidxj ,
ωμν ωμν = 2k4 ∗ ̟2,
Dν ωνμdxμ = k2Dˆj ̟jidxi − 2k4 ∗ ̟2Ωdt + 2k4 ∗ ̟2b.
Using (2.42) this leads to
with the Weylinvariant tensor
S = − 2
k
2
(Ωdt − b)2 ξij ξij + k4s − 5k6 (Ωdt − b)2 ∗ ̟2
s = 2 (Ωdt − b) dxiηjiDˆj ∗ ̟ + ∗̟2dℓ2 − Kˆ (Ωdt − b)2 .
In the derivative expansion (2.41), two explicit divergences appear at vanishing k.
The first originates from the first term of S, which is the shear contribution to the
Weylcovariant scalar curvature R of the threedimensional AdS boundary (eq. (4.3)).23 The
second divergence comes from the Cotton tensor and is also due to the shear. It is fortunate
— and expected — that counterterms coming from equalorder (nonexplicitly written) σ2
contributions, cancel out these singular terms. This is suggestive that (2.41) is wellbehaved
at zerok, showing that the reconstruction of Ricciflat spacetimes works starting from
twodimensional Carrollian fluid data.
We will not embark here in proving finiteness at k = 0, but rather confine our analysis
to situations without shear, as we discussed already in section 2.2 for Einstein spacetimes.
23This divergence is traced back in the GaussCodazzi equation relating the intrinsic and extrinsic
curvatures of an embedded surface, to the intrinsic curvature of the host. When the size of a fiber shrinks, the
extrinsiccurvature contribution diverges.
and reads:
Here
Vanishing σ in the pseudoRiemannian boundary I implies indeed vanishing ξij in the
Carrollian (see (4.2)), and in this case, the divergent terms in S and C are absent. Of course,
other divergences may occur from higherorder terms in the derivative expansion. To avoid
dealing with these issues, we will focus on the resummed version of (2.41) i.e. (2.53), valid
for algebraically special bulk geometries. This closed form is definitely smooth at zero k
dsr2es. flat = −2 (Ωdt − b) dr + rα +
dt
+ r2dℓ2 + s +
(8πGεr + c ∗ ̟) .
(Ωdt − b)2
ρ2
rθΩ
2
ρ2 = r2 + ∗̟2,
(4.9)
(4.10)
HJEP07(218)65
dℓ2, Ω, b = bidxi, α = αidxi, θ and ∗̟ are the Carrollian geometric objects introduced
earlier, while c and ε are the zerok (finite) limits of the corresponding relativistic functions.
Expression (4.9) will grant by construction an exact Ricciflat spacetime provided the
conditions under which (2.53) was Einstein are fulfilled in the zerok limit. These conditions
are the set of Carrollian hydrodynamic equations (3.49), (3.50), (3.51) and (3.52), and the
integrability conditions, as they emerge from (2.56) and (2.58) at vanishing k. Making the
latter explicit is the scope of next section.
Notice eventually that the Ricciflat line element (4.9) inherits Weyl invariance from
its relativistic ancestor. The set of transformations (3.24), (3.25) and (3.27), supplemented
with ∗̟ → B ∗ ̟, ε → B3ε and c → B3c, can indeed be absorbed by setting r → Br (s
is Weyl invariant), resulting thus in the invariance of (4.9). In the relativistic case this
invariance was due to the AdS conformal boundary. In the case at hand, this is rooted to
the location of the twodimensional spatial boundary S at null infinity I +.
4.2
Resummation of the Ricciflat derivative expansion
The Cotton tensor in Carrollian geometry.
The Cotton tensor monitors from the
boundary the global asymptotic structure of the bulk fourdimensional Einstein spacetime
(for higher dimensions, the boundary Weyl tensor is also involved, see footnote 11). In
order to proceed with our resummability analysis, we need to describe the zerok limit of
the Cotton tensor (2.32) and of its conservation equation (2.33).
As already mentioned, at vanishing k divergences do generally appear for some
components of the Cotton tensor. These divergences are no longer present when (2.54) is satisfied
(see footnote 23), i.e. in the absence of shear, which is precisely the assumption under
which we are working with (4.9). Every piece of the threedimensional relativistic Cotton
tensor appearing in (2.34) has thus a welldefined limit. We therefore introduce
χi = kli→m0 ci,
Xij = kli→m0 cij ,
1
1
ψi = lim
k→0 k2 (ci − χi) ,
Ψij = lim
k→0 k2 (cij − Xij ) .
(4.11)
(4.12)
The time components c0, c00 and c0i = ci0 vanish already at finite k (due to (2.36)),
and χi, ψi, Xij and Ψij are thus genuine Carrollian tensors transforming covariantly under
Finally, the weight1 symmetric and traceless ranktwo tensors read:
Observe that c and the subleading terms ψi and Ψij are present only when the vorticity is
nonvanishing (∗̟ 6= 0). All these are of gravitomagnetic nature.
The tensors c, χi, ψi, Xij and Ψij should be considered as the twodimensional
Carrollian resurgence of the threedimensional Riemannian Cotton tensor. They should be
referred to as Cotton descendants (there is no Cotton tensor in two dimensions anyway),
and obey identities inherited at zero k from its conservation equation.24 These are similar
to the hydrodynamic equations (3.49), (3.50), (3.51) and (3.52), satisfied by the different
pieces of the energymomentum tensor ε, Qi, πi, Σij and Ξij , and translating its
conservation. In the case at hand, the absence of shear trivializes (3.50) and discards the last term
in the other three equations:
c =
DˆlDˆl + 2Kˆ
∗ ̟.
χj =
21 ηlj DˆlKˆ +
12 Dˆj Aˆ − 2 ∗ ̟Rˆj ,
ψj = 3ηlj Dˆl ∗ ̟2.
Xij =
21 ηlj DˆlRˆi +
21 ηliDˆj Rˆl,
1
Ψij = DˆiDˆj ∗ ̟ − 2 aij DˆlDˆl ∗ ̟ − ηij Ω
1 Dˆt ∗ ̟2.
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
(4.18)
(4.19)
(4.20)
Carrollian diffeomorphisms. Actually, in the absence of shear the Cotton current and stress
tensor are given exactly (i.e. for finite k) by ci = χi + k2ψi and cij = Xij + k2Ψij .
The scalar c(t, x) is Weylcovariant of weight 3 (like the energy density). As expected,
it is expressed in terms of geometric Carrollian objects built on thirdderivatives of the
twodimensional metric dℓ2, bi and Ω:
Similarly, the forms χi and ψi, of weight 2, are
HJEP07(218)65
12 Dˆj c + 2χi̟ij +
Ω
1 Dˆtc + Dˆiχi = 0,
Ω
Ω
1 Dˆtψj − DˆiΨij = 0,
1 Dˆtχj − DˆiXij = 0.
One appreciates from these equations why it is important to keep the subleading corrections
at vanishing k, both in the Cotton current cμ and in the Cotton stress tensor cμν . As for the
energymomentum tensor, ignoring them would simply lead to wrong Carrollian dynamics.
24Observe that the Cotton tensor enters in eq. (2.60) with an extra factor k1 , the origin of which is
explained in footnote 9. Hence, the advisable prescription is to analyze the smallk limit of k1 ∇μCμν = 0.
The resummability conditions.
We are now ready to address the problem of
resummability in Carrollian framework, for Ricciflat spacetimes. In the relativistic case, where one
describes relativistic hydrodynamics on the pseudoRiemannian boundary of an
asymptotically locally AdS spacetime, resummability — or integrability — equations are eqs. (2.56)
and (2.58). These determine the friction components of the fluid energymomentum tensor
in terms of geometric data, captured by the Cotton tensor (current and stress components),
via a sort of gravitational electricmagnetic duality, transverse to the fluid congruence.
Equipped with those, the fluid equations (2.1) guarantee that the bulk is Einstein, i.e. that
bulk Einstein equations are satisfied.
Correspondingly, using (3.46), (3.47), (4.11) and (4.12), the zerok limit of eq. (2.56)
sets up a duality relationship among the Carrollianfluid heat current Qi and the
Carrolliangeometry thirdderivative vector χi:
Qi =
1
8πG
ηjiχj = − 16πG
1
DˆiKˆ
− ηjiDˆj Aˆ + 4 ∗ ̟ηjiRˆj ,
while eqs. (2.58) allow to relate the Carrollianfluid quantities Σij and Ξij , to the
Carrolliangeometry ones Xij and Ψij :
and
Σij =
1
8πG ηliXlj =
1
16πG
ηkj ηliDˆkRˆl − Dˆj Rˆi ,
Ξij =
1
8πG ηliΨlj =
1
8πG
ηliDˆlDˆj ∗ ̟ +
1
2 ηij DˆlDˆl ∗ ̟ − aij Ω
1 Dˆt ∗ ̟2 .
One readily shows that (3.48) is satisfied as a consequence of the symmetry and
tracelessness of Xij and Ψij .
One can finally recast the Carrollian hydrodynamic equations (3.49), (3.50), (3.51)
and (3.52) for the fluid under consideration. Recalling that the shear is assumed to vanish,
ξij =
1
2Ω
i
eq. (3.50) is trivialized. Furthermore, eq. (3.52) is automatically satisfied with Qj and Σ j
given above, thanks also to eq. (4.20). We are therefore left with two equations for the
energy density ε and the heat current πi:
• one scalar equation from (3.49):
• one vector equation from (3.51):
− Ω
1 Dˆtε +
1
16πG
Dˆi DˆiKˆ
− ηjiDˆj Aˆ + 4 ∗ ̟ηjiRˆj
= 0;
Dˆj ε + 4 ∗ ̟ηij Qi +
Ω
2 Dˆtπj − 2DˆiΞij = 0
with Qi and Ξij given in (4.21) and (4.23).
(4.21)
(4.22)
(4.23)
(4.24)
(4.25)
(4.26)
These last two equations are Carrollian equations, describing time and space evolution
of the fluid energy and heat current, as a consequence of transport phenomena like heat
conduction and friction. These phenomena have been identified by duality to geometric
quantities, and one recognizes distinct gravitoelectric (like Kˆ) and gravitomagnetic
contributions (like Aˆ). It should also be stressed that not all the terms are independent and
one can reshuffle them using identities relating the Carrollian curvature elements. In the
absence of shear, (3.23) holds and all information about Rˆij in (3.39) is stored in Kˆ and
Aˆ, while other geometrical data are supplied by Rˆi in (3.38). All these obey
Ω
1 DˆtKˆ
Ω
2 Dˆt ∗ ̟ + Aˆ = 0,
− aij DˆiRˆj = 0,
Ω
1 DˆtAˆ + ηij DˆiRˆj = 0,
(4.27)
χi and Xij .
which originate from threedimensional Riemannian Bianchi identities and emerge along
the ktozero limit.
Summarizing.
Our analysis of the zerok limit in the derivative expansion (2.53), valid
assuming the absence of shear, has the following salient features.
• As the general derivative expansion (2.41), this limit reveals a twodimensional spatial
boundary S located at I +
. It is endowed with a Carrollian geometry, encoded in
aij , bi and Ω, all functions of t and x. This is inherited from the conformal
threedimensional pseudoRiemannian boundary I of the original Einstein space.
• The Carrollian boundary S is the host of a Carrollian fluid, obtained as the limit
of a relativistic fluid, and described in terms of its energy density ε, and its friction
tensors Qi, πi, Σij and Ξij .
• When the friction tensors Qi, Σij and Ξij of the Carrollian fluid are given in terms
of the geometric objects χi, Xij and Ψij using (4.21), (4.22) and (4.23), and when
the energy density ε and the current πi obey the hydrodynamic equations (4.25)
and (4.26), the limiting resummed derivative expansion (4.9) is an exact Ricciflat
• The bulk spacetime is in general asymptotically locally flat. This property is encoded
in the zerok limit of the Cotton tensor, i.e. in the Cotton Carrollian descendants c,
The bulk Ricciflat spacetime obtained following the above procedure is algebraically
special. We indeed observe that the bulk congruence ∂r is null. Moreover, it is geodesic
and shearfree. To prove this last statement, we rewrite the metric (4.9) in terms of a null
tetrad (k, l, m, m¯):
dsr2es. flat = −2kl + 2mm¯ ,
k · l = −1 ,
m · m¯ = 1 ,
(4.28)
where k = − (Ωdt − b) is the dual of ∂r and
(here ψ = ψidxi), along with
l = −dr − rα −
rθΩ
2
dt +
6 ∗ ̟
Ωdt − b
2ρ2
8πGεr + c ∗ ̟ − ρ2Kˆ ,
Q
Using the above results and repeating the analysis of appendix A.2 in [13], we find that ∂r
is shearfree due to (4.24).
According to the GoldbergSachs theorem, the bulk spacetime (4.9) is therefore of
Petrov type II, III, D, N or O. The precise type is encoded in the Carrollian tensors ε ,
2mm¯ = ρ2dℓ2 .
ε
± = ε ± 8πG
c,
i
Q
i± = Qi ± 8πG χi,
Σi±j = Σij ± 8πG
Xij .
i
i
Q+ =
Σ+ =
i
i
4πG
4πG χζ dζ,
Xζζ dζ2,
(4.29)
(4.30)
±
(4.31)
(4.32)
(4.33)
Working again in holomorphic coordinates, we find the compact result
and their complexconjugates Q
− and Σ−. These Carrollian geometric tensors encompass
the information on the canonical complex functions describing the Weyltensor
decomposition in terms of principal null directions — usually referred to as Ψa, a = 0, . . . , 4.
5
Examples
There is a plethora of Carrollian fluids that can be studied. We will analyze here the class
of perfect conformal fluids, and will complete the discussion of section 3.2 on the Carrollian
RobinsonTrautman fluid. In each case, assuming the integrability conditions (4.21), (4.22)
and (4.23) are fulfilled and the hydrodynamic equations (4.25) and (4.26) are obeyed, a
Ricciflat spacetime is reconstructed from the Carrollian spatial boundary S at I +. More
examples exist like the Pleban´skiDemian´ski or the Weyl axisymmetric solutions, assuming
extra symmetries (but not necessarily stationarity) for a viscous Carrollian fluid. These
would require a more involved presentation.
5.1
Stationary Carrollian perfect fluids and Ricciflat KerrTaubNUT families
We would like to illustrate our findings and reconstruct from purely Carrollian fluid
dynamics the family of KerrTaubNUT stationary Ricciflat black holes. We pick for that
the following geometric data: aij (x), bi(x) and Ω = 1. Stationarity is implemented in these
fluids by requiring that all the quantities involved are time independent.
Under this assumption, the Carrollian shear ξij vanishes together with the
Carrollian expansion θ, whereas constant Ω makes the Carrollian acceleration ϕi vanish as well
(eq. (3.10)). Consequently
Aˆ = 0,
Rˆi = 0,
and we are left with nontrivial curvature and vorticity:
Kˆ = Kˆ = K,
̟ij = ∂[ibj] = ηij ∗ ̟.
The WeylCarroll spatial covariant derivative Dˆi reduces to the ordinary covariant
derivative ∇i, whereas the action of the WeylCarroll temporal covariant derivative Dˆt vanishes.
We further assume that the Carrollian fluid is perfect: Qi, πi, Σij and Ξij vanish.
This assumption is made according to the pattern of ref. [
10
], where the asymptotically
AdS KerrTaubNUT spacetimes were studied starting from relativistic perfect fluids. Due
to the duality relationships (4.21), (4.22) and (4.23) among the friction tensors of the
Carrollian fluid and the geometric quantities χi, Xij and Ψij , the latter must also vanish.
Using (4.14), (4.16) and (4.17), this sets the following simple geometric constraints:
(5.1)
(5.2)
(5.3)
(5.4)
(5.5)
(5.6)
(5.7)
HJEP07(218)65
and
χi = 0 ⇔ ∂iK = 0,
Ψij = 0 ⇔
1
∇i∇j − 2 aij ∇l∇l
∗ ̟ = 0,
whereas Xij vanishes identically without bringing any further restriction. These are
equations for the metric aij (x) and the scalar vorticity ∗̟, from which we can extract bi(x).
Using (4.13), we also learn that
where Δ = ∇l∇l is the ordinary Laplacian operator on S . The last piece of the geometrical
data, (4.15), it is nonvanishing and reads:
c = (Δ + 2K) ∗ ̟,
ψj = 3ηlj ∂l ∗ ̟2.
∂tε = 0,
∂iε = 0.
Finally, we must impose the fluid equations (4.25) and (4.26), leading to
The energy density ε of the Carrollian fluid is therefore a constant, which will be identified
to the bulk mass parameter M = 4πGε.
Every stationary Carrollian geometry encoded in aij (x) and bi(x) with constant scalar
curvature K hosts a conformal Carrollian perfect fluid with constant energy density, and
is associated with the following exact Ricciflat spacetime:
ds2perf. fl. = −2 (dt − b) dr +
2M r + c ∗ ̟ − Kρ2
ρ2
(dt − b)2 + (dt − b)
+ ρ2dℓ2, (5.8)
ψ
3 ∗ ̟
where ρ2 = r2 + ∗̟2. The vorticity ∗̟ is determined by eq. (5.4), solved on a
constantcurvature background.
Using holomorphic coordinates (see appendix A), a constantcurvature metric on S
dℓ2 =
K
2
ζζ¯,
2
∗ ̟ = n + a − P
2a
P
(1 − K) ζζ¯.
with
and
(5.9)
(5.10)
(5.11)
(5.12)
(5.13)
(5.14)
(5.15)
(5.16)
(5.17)
corresponding to S2 and E2 or H2 (sphere and Euclidean or hyperbolic planes). Using these
expressions we can integrate (5.4). The general solution depends on three real, arbitrary
HJEP07(218)65
parameters, n, a and ℓ:
The parameter ℓ is relevant in the flat case exclusively. We can further integrate (3.11)
and find thus
b =
i
P
n − P
a
ℓ
2P
(1 − K) ζζ¯
ζ¯dζ − ζdζ¯ .
It is straightforward to determine the last pieces entering the bulk resumed metric (5.8):
3 ∗ ̟
= 2ηji∂j ∗ ̟dxi = 2i
Ka + ℓ (1 − K) ζ¯dζ − ζdζ¯ .
P 2
In order to reach a more familiar form for the line element (5.8), it is convenient to
trade the complexconjugate coordinates ζ and ζ¯ for their modulus25 and argument
ζ = ZeiΦ,
with
ds2perf. fl. = − ρ2
and move from EddingtonFinkelstein to BoyerLindquist by setting
dt → dt −
r2 + (n − a)2
Δr
dr ,
dΦ → dΦ −
Ka + ℓ(1 − K) dr
Δr
Δr = −2M r + K r2 + a2 − n
2 + 2ℓ(n − a)(K − 1).
Δr
2ρ2
P 2
dt +
dZ2 +
2
P
2Z2
n − P
a
ℓ
2P
+
ρ
2
Δr
dr2
ρ2P 2 (Ka + ℓ (1 − K)) dt − r2 + (n − a)2 dΦ
2
(5.18)
0 < Θ < π for S2; Z = √R2 , 0 < R < +∞ for E2; Z = √
2 tanh Ψ2 , 0 < Ψ < +∞ for H2.
25The modulus and its range depend on the curvature. It is commonly expressed as: Z = √2 tan Θ2 ,
with
above.
P
.
(5.19)
This bulk metric is Ricciflat for any value of the parameters M , n, a and ℓ with K =
0, ±1. For vanishing n, a and ℓ, and with M > 0 and K = 1, one recovers the standard
asymptotically flat Schwarzschild solution with spherical horizon. For K = 0 or −1, this is
no longer Schwarzschild, but rather a metric belonging to the A class (see e.g. [83]). The
parameter a switches on rotation, while n is the standard nut charge. The parameter ℓ
is also a rotational parameter available only in the flatS case. Scanning over all these
parameters, in combination with the mass and K, we recover the whole KerrTaubNUT
family of black holes, plus other, less familiar configurations, like the Ametric quoted
For the solutions at hand, the only potentially nonvanishing Carrollian boundary
Cotton descendants are c and ψ, displayed in (5.13) and (5.14). The first is nonvanishing
for asymptotically locally flat spacetimes, and this requires nonzero n or ℓ. The second
measures the bulk twist. In every case the metric (5.18) is Petrov type D.
We would like to conclude the example of Carrollian conformal perfect fluids with a
comment regarding the isometries of the associated resummed Ricciflat spacetimes with
line element (5.18). For vanishing a and ℓ, there are four isometry generators and the
field is in this case a stationary gravitoelectric and/or gravitomagnetic monopole (mass
and nut parameters M , n). Constantr hypersurfaces are homogeneous spaces in this case.
The number of Killing fields is reduced to two (∂t and ∂Φ) whenever any of the rotational
parameters a or ℓ is nonzero. These parameters make the gravitational field dipolar.
The bulk isometries are generally inherited from the boundary symmetries, i.e. the
symmetries of the Carrollian geometry and the Carrollian fluid. The timelike Killing field
∂t is clearly rooted to the stationarity of the boundary data. The spacelike ones have legs
on ∂Φ and ∂Z , and are associated to further boundary symmetries. From a Riemannian
viewpoint, the metric (5.9) with (5.10) on the twodimensional boundary surface S admits
three Killing vector fields:
X 1 = i ζ∂ζ − ζ¯∂ζ¯ ,
X 2 = i
X 3 =
1 +
1 − 2
K ζ2 ∂ζ −
1 − 2
K ¯2 ∂ζ¯ ,
ζ
2
K ζ2 ∂ζ +
1 +
2
K ¯2 ∂¯,
ζ ζ
closing in so(3), e2 and so(2, 1) algebras for K = +1, 0 and −1 respectively. The Carrollian
structure is however richer as it hinges on the set {aij , bi, Ω}. Hence, not all Riemannian
isometries generated by a Killing field X of S are necessarily promoted to Carrollian
symmetries. For the latter, it is natural to further require the Carrollian vorticity be
invariant:
L
X ∗ ̟ = X (∗̟) = 0.
(5.20)
(5.21)
(5.22)
(5.23)
Condition (5.23) is fulfilled for all fields X A (A = 1, 2, 3) in (5.20), (5.21) and (5.22),
only as long as a = ℓ = 0, since ∗̟ = n. Otherwise ∗̟ is nonconstant and only X 1 =
= ∂Φ leaves it invariant. This is in line with the bulk isometry properties
discussed earlier, while it provides a Carrollianboundary manifestation of the rigidity
Vorticityfree Carrollian fluid and the Ricciflat RobinsonTrautman
The zerok limit of the relativistic RobinsonTrautman fluid presented in section 3.2
(eqs. (3.53)–(3.56)) is in agreement with the direct Carrollian approach of section 4.2.
Indeed, it is straightforward to check that the general formulas (4.13)–(4.17) give c = 0
∂ζ Kdζ − ∂ζ¯Kdζ¯ ,
X =
P 2 ∂ζ P 2∂t∂ζ ln P dζ2 − ∂ζ¯ P 2∂t∂ζ¯ ln P dζ¯2 ,
while ψi = 0 = Ψij . These expressions satisfy (4.18)–(4.20), and the duality relations (4.21),
(4.22) and (4.23) lead to the friction components of the energymomentum tensor Qi,
Σij and Ξij , precisely as they appear in (3.57), (3.58). The general hydrodynamic
equations (4.25), (4.26), are solved with26 πi = 0 and ε = ε(t) satisfying (3.59), i.e.
Robinson
Our goal is to present here the resummation of the derivative expansion (4.9) into
a Ricciflat spacetime dual to the fluid at hand. The basic feature of the latter is that
bi = 0 and Ω = 1, hence it is vorticityfree — on top of being shearless. With these data,
ds2RT = −2dt (dr + Hdt) + 2
2H = −2r∂t ln P + K −
r
2
with K = 2P 2∂ζ¯∂ζ ln P the Gaussian curvature of (3.53). This metric is Ricciflat provided
the energy density ε(t) = M4π(Gt) and the function P = P (t, ζ, ζ¯) satisfy (3.62). These are
algebraically special spacetimes of all types, as opposed to the KerrTaubNUT family
studied earlier (Schwarzschild solution is common to these two families). Furthermore
they never have twist (ψ = Ψ = 0) and are generically asymptotically locally but not
globally flat due to χ and X .
The specific Petrov type of RobinsonTrautman solutions is determined by analyzing
the tensors (4.31), or (4.32) and (4.33) in holomorphic coordinates:
(5.24)
(5.25)
(5.26)
ε
+ =
M (t)
4πG
,
1
Q+ = − 8πG ∂ζ Kdζ,
We find the following classification (see [12]):
II generic;
III with ε+ = 0 and ∇iQ+i = 0;
1
Σ+ = − 4πGP 2 ∂ζ P 2∂t∂ζ ln P dζ2.
(5.27)
26Since πi is not related to the geometry by duality as the other friction and heat tensors, it can a priori
assume any value. It is part of the Carrollian RobinsonTrautman fluid definition to set it to zero.
D with 2Qi+Qj+ = 3ε+Σi+j and vanishing traceless part of ∇(iQj+).
6
The main message of our work is that starting with the standard AdS holography, there is
a welldefined zerocosmologicalconstant limit that relates asymptotically flat spacetimes
to Carrollian fluids living on their null boundaries.
In order to unravel this relationship and make it operative for studying holographic
duals, we used the derivative expansion. Originally designed for asymptotically antide
Sitter spacetimes with cosmological constant Λ = −3k2, this expansion provides their line
element in terms of the conformal boundary data: a pseudoRiemannian metric and a
relativistic fluid. It is expressed in EddingtonFinkelstein coordinates, where the
zerok limit is unambiguous: it maps the pseudoRiemannian boundary I onto a Carrollian
geometry R × S , and the conformal relativistic fluid becomes Carrollian.
The emergence of the conformal Carrollian symmetry in the Ricciflat asymptotic is
not a surprise, as we have extensively discussed in the introduction. In particular, the
BMS group has been used for investigating the asymptotically flat dual dynamics. What is
remarkable is the efficiency of the derivative expansion to implement the limiting procedure
and deliver a genuine holographic relationship between Ricciflat spacetimes and conformal
Carrollian fluids. These are defined on S but their dynamics is rooted in R × S .
Even though proving that the derivative expansion is unconditionally wellbehaved in
the limit under consideration is still part of our agenda, we have demonstrated this property
in the instance where it is resummable.
The resummability of the derivative expansion has been studied in our earlier works
about antide Sitter fluid/gravity correspondence. It has two features:
• the shear of the fluid congruence vanishes;
• the heat current and the viscous stress tensor are determined from the Cotton current
and stress tensor components via a transverse (with respect to the velocity) duality.
The first considerably simplifies the expansion. Together with the second, it ultimately
dictates the structure of the bulk Weyl tensor, making the Einstein spacetime of special
Petrov type. The conservation of the energymomentum tensor is the only requirement left
for the bulk be Einstein. It involves the energy density (i.e. the only fluid observable left
undetermined) and various geometric data in the form of partial differential equations (as
is the RobinsonTrautman for the vorticityfree situation).
This pattern survives the zerok limit, taken in a frame where the relativistic fluid is
at rest. The corresponding Carrollian fluid — at rest by law — is required to be shearless,
but has otherwise acceleration, vorticity and expansion. Since the fluid is at rest, these are
geometric data, as are the descendants of the Cotton tensor used again to formulate the
duality that determines the dissipative components of the Carrollian fluid.
The study of the Cotton tensor and its Carrollian limit is central in our analysis. In
Carrollian geometry (conformal in the case under consideration) it opens the pandora box
of the classification of curvature tensors, which we have marginally discussed here. Our
observation is that the Cotton tensor grants the zerok limiting Carrollian geometry on S
with a scalar, two vectors and two symmetric, traceless tensors, satisfying a set of identities
inherited from the original conservation equation.
In a similar fashion, the relativistic energymomentum tensor descends in a scalar
(the energy density), two heat currents and two viscous stress tensors. This doubling is
suggested by that of the Cotton. The physics behind it is yet to be discovered, as it
requires a microscopic approach to Carrollian fluids, missing at present. Irrespective of
its microscopic origin, however, this is an essential result of our work, in contrast with
previous attempts. Not only we can state that the fluid holographically dual to a
Ricciflat spacetime is neither relativistic, nor Galilean, but we can also exhibit for the actually
Carrollian fluid the fundamental observables and the equations they obey.27 These are
quite convoluted, and whenever satisfied, the resummed metric is Ricciflat.
Our analysis, amply illustrated by two distinct examples departing from Carrollian
hydrodynamics and ending on widely used Ricciflat spacetimes, raises many questions,
which deserve a comprehensive survey.
As already acknowledged, the Cotton Carrollian descendants enter the holographic
reconstruction of a Ricciflat spacetime, along with the energymomentum data. It would
be rewarding to explore the information stored in these objects, which may carry the
boundary interpretation of the Bondi news tensor as well as of the asymptotic charges one
can extract from the latter.
We should stress at this point that Cotton and energymomentum data (and the
charges they transport) play dual rˆoles. The nut and the mass provide the best paradigm
of this statement. Altogether they raise the question on the thermodynamic interpretation
of magnetic charges. Although we cannot propose a definite answer to this question, the
tools of fluid/gravity holography (either AdS or flat) may turn helpful. This is tangible
in the case of algebraically special Einstein solutions, where the underlying integrability
conditions set a deep relationship between geometry and energymomentum i.e. between
geometry and local thermodynamics. To make this statement more concrete, observe the
heat current as constructed using the integrability conditions, eq. (4.21):
Qi = − 16πG
1
DˆiKˆ
− ηjiDˆj Aˆ + 4 ∗ ̟ηjiRˆj .
In the absence of magnetic charges, only the first term is present and it is tempting to
set a relationship between the temperature and the gravitoelectric curvature scalar Kˆ.
This was precisely discussed in the AdS framework when studying the RobinsonTrautman
relativistic fluid, in ref. [66]. Magnetic charges switch on the other terms, exhibiting natural
thermodynamic potentials, again related with curvature components (Aˆ and Rˆj ).
27From this perspective, trying to design fourdimensional flat holography using twodimensional
conformal field theory described in terms of a conserved twodimensional energymomentum tensor [42–44] looks
inappropriate.
We would like to conclude with a remark. On the one hand, we have shown that
the boundary fluids holographically dual to Ricciflat spacetimes are of Carrollian nature.
On the other hand, the stretched horizon in the membrane paradigm seems to be rather
described in terms of Galilean hydrodynamics [17, 18, 84]. Whether and how these two
pictures could been related is certainly worth refining.
Acknowledgments
We would like to thank G. Barnich, G. Bossard, A. Campoleoni, S. Mahapatra, O. Miskovic,
A. Mukhopadhyay, R. Olea and P. Tripathy for valuable scientific exchanges.
Marios
Petropoulos would like to thank N. Banerjee for the Indian Strings Meeting, Pune, India,
December 2016, P. Sundell, O. Miskovic and R. Olea for the Primer Workshop de Geometr´ıa
y F´ısica, San Pedro de Atacama, Chile, May 2017, and A. Sagnotti for the Workshop on
Future of Fundamental Physics (within the 6th International Conference on New Frontiers
in Physics — ICNFP), Kolybari, Greece, August 2017, where many stimulating discussions
on the topic of this work helped making progress. We thank each others home institutions
for hospitality and financial support. This work was supported by the ANR16CE310004
contract BlackdSString.
A
Carrollian boundary geometry in holomorphic coordinates
Using Carrollian diffeomorphisms (3.2), the metric (3.1) of the Carrollian geometry on the
twodimensional surface S can be recast in conformally flat form,
ηζζ¯ = − P 2
.
i
∗̟ =
P 2 dζdζ¯
with P = P (t, ζ, ζ¯) a real function, under the necessary and sufficient condition that the
Carrollian shear ξij displayed in (3.14) 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(t, ζ, ζ¯) and
b = bζ (t, ζ, ζ¯) dζ + bζ¯(t, ζ, ζ¯) dζ¯
the relativistic boundary (see footnote 13) with aζζ¯ = P12 is28
with bζ¯(t, ζ, ζ¯) = ¯bζ (t, ζ, ζ¯). Our choice of orientation is inherited from the one adopted for
The firstderivative Carrollian tensors are the acceleration (3.10), the expansion (3.14)
and the scalar vorticity (3.20):
ϕζ = ∂t
bζ + ∂ˆζ ln Ω,
Ω
2
θ = − Ω ∂t ln P,
ϕζ¯ = ∂t
ζ + ∂ˆζ¯ ln Ω,
iΩP 2
2
∂ζ Ω − ∂ζ¯ Ω
ˆ bζ
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
with
and we also quote:
Curvature scalars and vector are secondderivative (see (3.19), (3.22)):
∂ = ∂ +
ζ ζ
b
ζ
Ω
∂ = ∂ +
¯ ¯
ζ ζ
∂ .
t
K = P
2
ˆ ˆ
∂ ∂ + ∂ ∂
ζ¯ ζ
ζ ζ¯
ˆ ˆ
ln P,
A = iP
2
ˆ ˆ
∂ ∂
ζ¯ ζ −
ˆ ˆ
∂ ∂
ζ ζ¯
rˆ =
ζ
2
Ω
∂ ln P ,
t
rˆ =
¯
ζ
2
¯
ζ
∂ ln P ,
t
ln P,
ϕ = iP
2
∂ ϕ
ζ ¯
ζ − ζ
∂ ϕ
¯ ζ
b
¯
ζ
Ω
,
b
ζ
Ω
= P
2
ζ t
¯ t
ζ ζ¯
ζ¯ ζ
ln Ω .
b
¯
ζ
Ω
Ω
2
2
Regarding conformal Carrollian tensors we remind the weight2 curvature
scalars (3.40):
and the weight1 curvature oneform (3.38):
Kˆ
= K +
ϕ ,
Aˆ
= A
Ω
Rˆ
ζ
∂ ϕ
t ζ −
2
¯
ζ
Ω
∂ ϕ
t ¯
ζ −
∂ + ϕ
¯
ζ
¯
ζ
The threederivative Cotton descendants displayed in (4.13)–(4.17) are a scalar
(A.7)
(A.9)
(A.10)
(A.11)
(A.12)
(A.13)
(A.14)
(A.15)
(A.16)
(A.17)
HJEP07(218)65
c =
Dˆ Dˆl
l
+ 2
Kˆ
∗
of weight 3 ( ̟ is of weght 1), two vectors
χ =
ζ
ψ = 3i
ζ
2
Dˆ Kˆ
ζ
Dˆ
ζ ∗
2
̟ ,
ζ
,
Dˆ Kˆ
¯
ζ
Dˆ Aˆ
¯
ζ
2
¯
ζ
,
χ =
¯
ζ
ψ =
¯
ζ
i
2
3i
Dˆ
¯
ζ ∗
2
̟ ,
of weight 2, and two symmetric and traceless tensors
X
Ψ
ζζ
ζζ
= i
Dˆ Rˆ
ζ ζ
,
Dˆ Dˆ
ζ ζ ∗
Ψ
¯¯
ζζ
¯¯
ζζ
i
Dˆ Rˆ
¯ ¯
ζ ζ
,
Dˆ Dˆ
¯ ¯
ζ
ζ ∗
̟,
of weight 1. Notice that in holomorphic coordinates a symmetric and traceless tensor S
ij
has only diagonal entries: S
¯
ζζ
We also quote for completeness (useful e.g. in eq. (A.11)):
ˆ
K = K + P
2
ζ
b¯
ζ
Ω
b
ζ
Ω
bζbζ¯
Ω
2
b¯
ζ
Ω
∂ + 2 ∂¯ + 2
ζ ζ
∂ ∂ ln P
t t
b
ζ
Ω
bζbζ¯
Ω
2
2
with K = 2P ∂ζ¯∂ζ ln P the ordinary Gaussian curvature of the twodimensional metric (A.1).
We also remind for convenience some expressions for the determination of WeylCarroll
covariant derivatives. If Φ is a weightw scalar function
Dˆζ Φ = ∂ˆζ Φ + wϕζ Φ,
Dˆζ¯Φ = ∂ˆζ¯Φ + wϕζ¯Φ.
For weightw form components Vζ and Vζ¯ the WeylCarroll derivatives read:
Dˆζ Vζ = ∇ˆ ζ Vζ + (w + 2)ϕζ Vζ ,
Dˆζ Vζ¯ = ∇ˆ ζ Vζ¯ + wϕζ Vζ¯,
Dˆζ¯Vζ¯ = ∇ˆ ζ¯Vζ¯ + (w + 2)ϕζ¯Vζ¯,
Dˆζ¯Vζ = ∇ˆ ζ¯Vζ + wϕζ¯Vζ ,
while the Carrollian covariant derivatives are simply:
∇ˆ ζ Vζ =
1 ˆ
P 2 ∂ζ P 2Vζ ,
∇ˆ ζ Vζ¯ = ∂ˆζ Vζ¯,
∇ˆ ζ¯Vζ¯ =
∇ˆ ζ¯Vζ = ∂ˆζ¯Vζ .
1 ˆ
P 2 ∂ζ¯ P 2Vζ¯ ,
(A.18)
(A.20)
(A.21)
(A.22)
(A.23)
Finally,
Dˆ DˆkΦ = P 2 ∂ˆζ ∂ˆζ¯Φ + ∂ˆζ¯∂ˆζ Φ + wΦ ∂ˆζ ϕζ¯ + ∂ˆζ¯ϕζ
k
+ 2w ϕζ ∂ˆζ¯Φ + ϕζ¯∂ˆζ Φ + wϕζ ϕζ¯Φ
Attribution License (CCBY 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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