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Carrollian Fluids and Spontaneous Breaking of Boost Symmetry
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Carrollian Fluids and Spontaneous Breaking of Boost Symmetry
Jay Armas1,2,and Emil Have3,4,?
1Institute for Theoretical Physics and Dutch Institute for Emergent Phenomena, University of Amsterdam,
1090 GL Amsterdam, The Netherlands
2Institute for Advanced Study, University of Amsterdam, Oude Turfmarkt 147, 1012 GC Amsterdam, The Netherlands
3School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh,
Peter Guthrie Tait road, Edinburgh EH9 3FD, United Kingdom
4Niels Bohr International Academy, Niels Bohr Institute, University of Copenhagen,
Blegdamsvej 17, DK-2100 Copenhagen ?, Denmark
(Received 28 September 2023; revised 16 January 2024; accepted 29 February 2024; published 17 April 2024)
In the hydrodynamic regime, field theories typically have their boost symmetry spontaneously broken
due to the presence of a thermal rest frame although the associated Goldstone field does not acquireindependent dynamics. We show that this is not the case for Carrollian field theories where the boost
Goldstone field plays a central role. This allows us to give a first-principles derivation of the equilibrium
currents and dissipative effects of Carrollian fluids. We also demonstrate that the limit of vanishing speed oflight of relativistic fluids is a special case of this class of Carrollian fluids. Our results shine light on the
thermodynamic properties and thermal partition functions of Carrollian field theories.
DOI: 10.1103/PhysRevLett.132.161606
Introduction. —In the past few years Carrollian physics,
emerging by taking the limit of the vanishing speed of light,
has been found useful for describing a variety of phenomena
in contexts ranging from black holes [1–4], cosmology [5,6],
gravity [3,7–16], to hydrodynamics [4,5,17 –24]. Concretely,
Carrollian fluids can be used to describe Bjorken flow, which
is relevant for models of the quark-gluon plasma, cf. [23]
(and its conformal generalization, Gubser flow [25]).
Carrollian fluids also model dark energy in inflationary
models [5]. Furthermore, Carrollian symmetries are expected
to have a role to play in exotic phases of matter (e.g., viaCarroll-fracton dualities [26–29]and in superconducting
twisted bilayer graphene [30]).
Many of the properties encountered in this Carrollian
limit are expected to be explained from underlying
quantum field theories with inherent Carrollian sym-metries. Indeed, if conformal symmetry is present in
addition, such theories would be putative holographic
duals to flat space gravity [31–41].H o w e v e r ,w h e n
attempting to formulate such Carrollian field theories,
several issues have been pointed out including violations
of causality, lack of well-defined thermodynamics, and ill-defined partition functions [5,24] . Our goal is to show that
the lack of well-defined thermodynamics in Carrollian
field theories is expected in the hydrodynamic regime, butthat this issue can be cured when carefully accounting for
the Carrollian symmetries.
The approach we take is to consider the hydrodynamic
regime of such putative Carrollian field theories and showhow to construct their equilibrium partition function andnear-equilibrium dynamics. In particular, we will show thatthere is no proper notion of temperature in Carrollian fluids
unless the Goldstone field of spontaneous broken boost
symmetry is taken into account. This allows us to constructa well-defined hydrodynamic theory of Carrollian fluids(similar to framids in the language of [42,43] ). In the
process, we show that seemingly different approaches toCarrollian hydrodynamics previously pursued in the liter-ature [5,17 –24]are in fact equivalent and special cases of
the Carrollian fluids we derive.
The fact that boost symmetry is spontaneously broken
in hydrodynamics is not unexpected. Thermal states
break the boost symmetry spontaneously due to thepresence of a preferred rest frame aligned with the thermal
vector [43,44] . In the context of hydrodynamics the
thermal vector is the combination u
μ=Tof the unit
normalized fluid velocity uμand temperature T. On the
other hand, the Goldstone field associated with the break-
ing of boost symmetry does not typically feature in the
low energy spectrum of the theory because it is determinedin terms of the other dynamical fields (see, e.g., [42]). This
is easy to show for relativistic fluids. Consider a ( d?1)-
dimensional spacetime metric g
μν?EAμEμ
Awhere EAμis the
set of vielbeins, μ?0;…;dare spacetime indices, and
A?0;…;dare internal Lorentz indices. The Goldstone
field associated to spontaneous breaking of Lorentz boostPublished by the American Physical Society under the terms of
theCreative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article ’s title, journal citation,
and DOI. Funded by SCOAP3.PHYSICAL REVIEW LETTERS 132, 161606 (2024)
0031-9007 =24=132(16) =161606(7) 161606-1 Published by the American Physical Societysymmetry is the vector lA?Λ0
Awhere Λ0
Ais the Lorentz
boost matrix [42], and acquires an expectation value
hlAi?δ0
Ain the ground state. In thermal equilibrium
we can construct an equilibrium partition function S?Rdd?1x???????gpP?T;uμlμ?where Pis the fluid pressure. The
temperature is given by T?T0=jKjwith T0a global
constant temperature, and the fluid velocity is uμ?
Kμ=jKjsatisfying uμuμ??1and defined in terms of
the Killing (thermal) vector Kμwith modulus jKj2?
?gμνKμKν. We furthermore defined lμ?EAμlAsatisfying
lμlμ??1. The Goldstone equation of motion obtained
from varying the partition function Swith respect to
lAis[42]
?δμ
ν?lμlν?δS
δlμ?0: ?1?
This equation implies uμ??lνuνlμfor arbitrary thermo-
dynamic coefficients, and can only be satisfied if lμ?uμ.
Indeed, we see that the dynamics of the Goldstone field lμis
determined by the dynamics of the fluid velocity and hencecan be removed from the hydrodynamic description. Thesame conclusion is reached for the case of spontaneousbreaking of Galilean (Bargmann) boost symmetry [45](see
also[44]). However, in the case of Carrollian symmetry, as
we will show, the boost Goldstone acquires its ownindependent dynamics. Below we introduce Carrolliangeometry and use it to show that, naively, there is nowell-defined notion of temperature.
Carrollian geometry and the lack of temperature. —A
weak Carrollian geometry on a ( d?1)-dimensional mani-
fold Mis defined by a Carrollian structure ?v
μ;hμν?
consisting of the nowhere-vanishing Carrollian vector field
vμ, and the corank-1 symmetric tensor hμν, the “ruler,”
satisfying hμνvν?0. It is useful to define inverses ?τμ;hμν?
satisfying vμτμ??1, and τμhμν?0, as well as the
completeness relation ?vμτν?hμρhρν?δμ
ν. It is also
useful to introduce the spatial vielbeins eaμand their
inverses eμ
awhich can be used to write hμν?eaμeaν.
Under Carrollian boosts, the inverses transform as
δCτμ?λμ;δChμν?2λρhρ?μvν?; ?2?
corresponding to δCeμ
a?vμλa, where we have used
thatvμλμ?0.A strong Carrollian geometry is a weak
Carrollian geometry together with an affine connection.
We focus mainly on weak Carrollian geometry but discussstrong Carrollian geometry in Appendix D of the Supple-mental Material [46].
Given the Carroll geometry and the existence of a
thermal vector in equilibrium, namely the spacetimeKilling vector k
μ, one can proceed as for other (non)-
Lorentzian field theories [54–65]and construct an equi-
librium partition function by identifying the invariantscalars under all local symmetries which the pressure
Pcan depend on, as above Eq. (1)for the relativistic case.
A more thorough construction of the partition function will
be given in a later section. Here we note that for non-relativistic theories the temperature Tis given by the scalar
T?T
0=?kμτμ?with T0a constant global temperature.
However, in the Carrollian case Tis not invariant under
boost transformations since δC?kμτμ??kμλμ≠0. Indeed,
there is no well-defined notion of temperature for arbitrary
observers [66]. This is rooted in the fact that it is not
possible to impose a timelike normalization condition onspacetime vectors such as the fluid velocity since u
μτμis
not boost invariant. This argument does not rely on any
specific model of Carrollian (quantum) field theory sincethese statements are valid in the hydrodynamic regime.This suggests that well-defined thermodynamic limits ofsuch putative theories are subtle. Below we introduce the
boost Goldstone and use it to show that it can be used to
define an appropriate notion of temperature.
The Carroll boost Goldstone. —We define the boost
Goldstone as the vector θ
μwhich transforms under
Carrollian boosts λμas
δCθμ??hμνλν; ?3?
where λμvμ?0. This implies that only the spatial part of θμ
is physical, which we can enforce by endowing the
Goldstone with a timelike Stueckelberg symmetry of theform
δ
Sθμ?χvμ; ?4?
where χis an arbitrary function [67]. With this we can build
the boost and Stueckelberg invariant vielbeine
?τμ?τμ?hμνθν; ?eμ
a?eμ
a?vμθμeμa; ?5?
which lead to the following invariant ruler:
?hμν?δab?eμ
a?eν
b?hμν?vμvν?θ2?2τρθρ??2v?μθν?;?6?
where θ2?hμνθμθν. Together with vμandhμν, these form
anAristotelian structure [68],
?hμν?τν?0;vμ?τμ??1; ?hμρhρν?δμ
ν?vμ?τν??hμ
ν;?7?
that is partly dynamical due to the Goldstone θμ. The low-
energy effective action for the Carrollian boost Goldstone isa two-derivative Ho ˇrava–Lifshitz type action as we show in
Appendix A of [46]. If coupled to Carrollian gravity the
resultant action would be derivable from the limit of the
vanishing speed of light of the Einstein-Aether theory [69].
Before showing how the Goldstone allows one to define anotion of temperature, we first discuss the currents andconservation laws.PHYSICAL REVIEW LETTERS 132, 161606 (2024)
161606-2Currents and conservation laws. —We now consider an
arbitrary fluid functional (or free energy) S?τμ;hμν;θν/C138for a
Carrollian fluid with spontaneously broken boosts. The
variation of this functional is
δS?Z
dd?1xeh
?Tμδτμ?1
2Tμνδhμν?Kμδθμi
;?8?
where Tμis the energy current, Tμνthe stress-momentum
tensor, and Kμthe response to the Goldstone field. In
particular Kμ?0gives the analogous equation of motion
for the Goldstone as in (1). The measure is defined as
e?det?τμ;eaμ??det??τμ;eaμ?. The Ward identities for the
Stueckelberg and boost symmetries are, respectively,
vμKμ?0;Tνhνμ?Kμ: ?9?
The equation of motion for the Goldstone, Kμ?0, imposes
the condition Tνhνμ?0. In other words, the boost Ward
identity now becomes the equation of motion for the
Goldstone. The momentum-stress tensor is not boost invari-ant. In fact, computing the second variation, which mustvanish δ
C?δCS?? 0, we find that δCTμ?δCKμ?0and
δCTμν?2T?μhν?ρλρ. The associated energy-momentum
tensor (EMT) Tμ
ν??τνTμ?Tμρhρνis also not boost
invariant and transforms as
δCTμ
ν?Kνhμρλρ; ?10?
where we used (9). Doing the same for the Stueckelberg
symmetry, the condition δS?δSS?? 0implies that δSTμ
ν?
?χvμKν. Hence, the EMT is both boost and Stueckelberg
invariant if Kμ?0. The diffeomorphism Ward identity
reads as
e?1?μ?eTμ
ρ??Tμ?ρτμ?1
2Tμν?ρhμν?0; ?11?
wherewe used that Kμ?0. It is possible to obtain manifestly
boost invariant currents, including the EMT, by formulating
the action in terms of the effective Aristotelian structure (5),
as we show in Appendix B of [46].
Equilibrium partition function and Carrollian fluids. —
To derive the currents of Carrollian fluids, we consider the
equilibrium partition function construction. An equilibriumCarrollian background consists of a set of symmetry
parameters K??k
μ;λμ
K;χK?, where kμis a Killing vector
and λμ
Kis a boost symmetry parameter, while χKis a
Stueckelberg symmetry parameter. The various structures
transform according toδKvμ??kvμ?0;δKτμ??kτμ?λKμ?0;
δKhμν??khμν?0;
δKhμν??khμν?2λKρhρ?μvν??0;
δKθμ??kθμ?hμνλKμ?χKvμ?0: ?12?
The boost and Stueckelberg symmetry parameters trans-
form as
δλKμ??ξλKμ??kλμ;δχK??ξχK??kχ; ?13?
under infinitesimal diffeomorphisms generated by ξμ,
infinitesimal Carrollian boosts λμand Stueckelberg trans-
formations χ. As we show in Appendix B of [46],λKμand
χKwill not play a role in the effective fluid description.
Before enumerating the possible invariant scalars, we
must provide a gradient ordering. As usual we take the
geometry itself to be of ideal order, that is, τμ?hμν?
vμ?hμν?O?1?. Since θμenters the definition of ?τμit must
have the same ordering, θμ?O?1?. Gradients of these
structures are O???and hence suppressed in a hydrodynamic
expansion. Given this gradient scheme the only two idealorder invariants are
T?T
0=?τμkμ; ?u2?hμνuμuν; ?14?
where uμ?kμ=?τρkρ, which satisfies ?τμuμ?1. We note
that we can now define a notion of temperature Tthat is
invariant for all observers. The scalar ?u2is the modulus of
the spatial fluid velocity. Generically the fluid velocity can
be decomposed as uμ??vμ? ?uμ, where ?uμ??hμ
νuνwith
?hμ
ν? ?hμρhρν. We furthermore define ?uμ?hμνuν?hμν ?uν,
such that ?uμ??hμ
νuν,b u t ?uμ≠hμν?uν. Note in particular
thatuμdecomposes as follows relative to the Carrollian
structure:
uμ??vμ?1?θν?uν??hμ
νuν: ?15?
The hydrostatic partition function at ideal order is given by
S?Rdd?1xeP?T; ?u2?. Using the general actionvariation (8)
together with the “variational calculi ”δhμν?2v?μhν?ρδτρ?
hμρhνσδhρσandδvμ?vμvνδτν?hμνvρδhρνwe obtain the
ideal order currents:
Tμ
?0??Pvμ?sTuμ?m ?u2uμ;
Tμν
?0??Phμν?muμuν?2?sT?m ?u2?u?μθν?;
K?0?μ??sT?m ?u2? ?uμ; ?16?
where the subscript (0) indicates that the currents are of ideal
orderO?1?and the entropy sand mass density mare defined
viadP?sdT ?md ?u2. The associated EMT is given byPHYSICAL REVIEW LETTERS 132, 161606 (2024)
161606-3Tμ
?0?ν?Pδμ
ν?muμ?uν??sT?m ?u2??uμ?τν?θμ?uν?;?17?
which transforms as in (10).
The equation of motion for the Goldstone K?0?μ?0,
which is equivalent to the boost Ward identity, gives a
constraint on the dynamics
?sT?m ?u2? ?uμ?0; ?18?
and can be viewed as a framid condition for Carrollian fluids.
Defining the energy density as E??τμTμ
?0?, the Goldstone
equation has two branches of solutions: either E?P?
sT?m ?u2?0or ?uμ?0. Neither of them allows for the
elimination of the Goldstone θμfrom the low-energy
description. The constraint (18)was derived in equilibrium,
but we show in Appendix B of [46] that it also holds off
equilibrium, although it receives corrections due to dissipa-
tive effects. As such, together with (11), it provides the ideal
order dynamics for Carrollian fluids. Equations (16)–(18)are
a central result of this work as they provide a well-defined
notion of Carrollian fluids. Below we show that the c→0
limit of relativistic fluids gives rise to a Carrollian fluid
with ?uμ?0.
Thec→0limit of a relativistic fluid. —Thec→0limit
of relativistic fluids was considered in [17,18,20] (see
also [23]) for a specific class of metrics. The same limit
was taken in [24], where it was referred to as a “timelike
fluid.”Here, we demonstrate that these notions coincide and
correspond to the special case of the Carrollian fluid we
introduced above with ?uμ?0, and that the emergence of the
Goldstone can be understood from the ultralocal expansion
of the Lorentzian geometry. The relativistic EMT is given by
Tμν??E??P
c2UμUν??Pδμ
ν; ?19?
where UμUνgμν??c2, and where the “hat”indicates
relativistic thermodynamic quantities. To take the limit,
we first consider the metric and its inverse in “pre-ultra-
local (PUL) variables ”[3]
gμν??c2TμTν?Πμν;gμν??1
c2VμVν?Πμν;?20?
where TμVμ??1,TμΠμν?VμΠμν?0,ΠμρΠρν?
δμ
ν?VμTν. The leading order components of the PUL
variables correspond to the fields that make up the
Carrollian structure, e.g., Vμ?vμ?O?c2?. We write the
expansion of the relativistic fluid velocity relative to the PUL
variables as
Uμ??Vμ?c2uμ;
for some uμ. Crucially, Uμis invariant under local Lorentz
boosts while δCVμ?c2hμνλν?O?c4?, implying thatδCuμ??hμνλν?O?c2?. This shows that uμcannot be
identified with a fluid velocity in the Carrollian limit.Indeed, we may identify the spatial part of the leading order
term in the c
2expansion of uμwith the spatial part of the
boost Goldstone
Πμνuν?hμνθν?O?c2?? ?θμ?O?c2?: ?21?
Using this, together with Uμ?c2?τμ?O?c4?the EMT
becomes
Tμ
ν??E?P?vμ?τν?Pδμ
ν?O?c2?; ?22?
where EandPare the leading order contributions of ?Eand ?P,
respectively, satisfying the Euler relation E?P?sT.T h i s
is exactly the “timelike ”fluid of [24], corresponding to
the ?uμ?0branch of the Carrollian fluid we described
above [70].
It is instructive to take the c→0limit of the relativistic
equation of motion b?μTμ
ν?0, where b?is the Levi-Civita
connection of the spacetime metric gμν. Deferring the
details to Appendix C of [46], we note here that the
equations of motion in the limit c→0can be expressed as
vμ?μE??E?P?K;
hμν?νP???φμ?E?P???E?P?Khμν ?θν
?hμνvρe?ρ??E?P? ?θν/C138; ?23?
where K?hμνKμν??1
2hμν?vhμνis the trace of the
intrinsic torsion of the Carrollian structure [71], and is
sometimes referred to as the “Carrollian expansion, ”while
?φλ?2hλμvν??ντμ/C138?hλμhνσ ?θσKμν. The first term in ?φλis
sometimes referred to as the “Carrollian acceleration, ”while
the second term comes from the c2expansion of the Levi-
Civita connection. The covariant derivative e?is a Carroll
compatible connection that arises in the O?1?piece of the c2
expansion of the Levi-Civita connection, which we discuss
further in Appendix C of [46]. These equations are fully
covariant and reduce to the special case of the equations of
motion obtained in [17,18,20,23] when restricted to space-
time metrics that admit a Randers-Papapetrou parametriza-tion. Furthermore, these equations can be obtained byprojecting the conservation law (11) along the time and
spatial directions using (17) with ?u
μ?0. We thus have
shown that the “timelike ”fluid of [24] is the same as the
Carrollian fluid of [17,18,23] , and both are a special case of
the Carrollian fluid derived here.
Dissipation and modes. —In Appendix B of [46] we
show that at order O???the class of Carrollian fluids we
introduced is characterized by two hydrostatic coefficientsand ten dissipative coefficients. Here we study the effect of
specific coefficients in the linear spectrum of fluctuations.
We consider flat Carrollian space with τ
μ?δtμ,vμ??δμ
t,PHYSICAL REVIEW LETTERS 132, 161606 (2024)
161606-4hμν?δiμδiν, and hμν?δμ
iδν
i(see Appendix B of [46] for
more details). We then fluctuate the conservation equa-
tions (11) and the boost Ward identity (9)around an
equilibrium state with constant temperature T0, fluid
velocity vi
0, and Goldstone field θi
0, such that, e.g.,
θi?θi
0?δθi. Using plane wave perturbations with fre-
quency ωand wave vector ?kwe find a distinguishing
feature of these Carrollian fluids: the mode structurestrongly depends on whether the equilibrium state carries
nonzero velocity v
i
0.I fvi
0?0, the linearized equations only
admit a nontrivial solution if θi
0≠0,T0?0, and δvi?0.
Denoting the angle between the momentum kiandθi
0by?,
this leads to a single linear mode:
ω??1
j ?θ0jcos?j ?kj; ?24?
where we assumed that the value of the entropy density sin
equilibrium s0remains finite and nonvanishing when
T0→0; otherwise there is no mode. Interestingly, this
mode is not affected by any of the 12 transport coefficientsentering at order O???[72]. This spectrum corresponds to
the branch of solutions with ?u
μ?0, and hence it is the
expected spectrum arising from the c→0limit of an ideal
relativistic fluid.
On the other hand if vi
0≠0butθi
0?0a more interesting
spectrum can be obtained. For simplicity we only consider
the effect of a bulk viscosity s3and one anisotropic
viscosity s2. Besides a gapped mode, we find for d?2
a single diffusive mode of the form
ω?vi
0ki??iΓ1
2?εijvi
0kj?2; ?25?
where εijis the two-dimensional Levi-Civita symbol, and
Γ1?s2;0?T2
0χTT?jv0j2?3T0χTu?2jv0j2χuu?/C138
s0T0?T2
0χTT?jv0j2?2T0χTu?jv0j2χuu?/C138
?2s3;0
s0T0;
where s3;0is thevalue of s3in equilibrium, ditto s2,a n dw h e r e
we defined χTT???2P=?T?T?0,χuu???2P=? ?u2? ?u2?0, and
χTu???2P=?T? ?u2?0. The left-hand side of (25)is character-
istic of a fluid without boost symmetry [63]while the right-
hand side is typical of a diffusive mode. A salient signature ofCarrollian fluids is that the spectrum is only nontrivial for
states with nonzero equilibrium velocity v
i
0.
Discussion. —We have given a first-principles derivation
of Carrollian fluids based on symmetries, showing that thespontaneous breaking of boost symmetry plays a crucial
role in defining equilibrium partition functions of Carrollianfield theories in the hydrodynamic regime. It is interesting
to speculate whether this peculiar feature of Carrollianhydrodynamics can shed light on how to construct well-
defined partition functions using specific microscopic mod-
els of Carrollian field theories along the lines of [24,44] .
Different approaches to Carrollian hydrodynamics have ap-
peared in the literature in the past few years [5,17 –19,21 –24].
Revisiting the c→0limit of relativistic fluids we showed
that there are subtleties regarding the interpretation of the
dynamical variables that appear in the limit of the equations
of motion. In particular, we showed that what naivelyappeared to be a fluid velocity was in fact a Goldstone fieldassociated to the spontaneous breaking of boost symmetry.This allowed us to show that the different approaches are not
only equivalent to each other but also special cases of the
more general Carrollian fluids we introduced here. Webelieve it could be interesting to revisit the black holemembrane paradigm [2]in light of this new understanding.
The effective field theory geometry becomes Aristotelian
once taking the Goldstone field into account. This allowed
us to easily understand the dissipative structure of such
fluids using earlier results [63,73] . The spectrum of exci-
tations for certain classes of equilibrium states shares certainsimilarities with the spectrum of excitations of p-wave
fracton superfluids in which the Goldstone field associated
to the spontaneous breaking of dipole symmetry plays an
analogous role to the boost Goldstone field [45,64,65] , albeit
in the Carroll case a nonvanishing fluid velocity is needed.We believe that this relation can be made clearer if weconsider strong Carrollian geometries as we describe in
Appendix D of [46].
Finally, it would be interesting to consider the addition of
conformal symmetry as this could shine light on thermo-
dynamic properties of holographic dual theories of flat space
gravity. If we impose this symmetry the equation of state for
the branch ?u
μ?0becomes E?dPwhile for the branch
sT?m ?u2?0it imposes the relation m????d?1?= ?u2/C138P.
In addition, the number of first order transport coefficients
reduces from 12 to 8 (see Appendix B of [46]). It would be
interesting to consider this case in further detail and explore
its connections to flat space holography.
We are grateful to Arjun Bagchi, Jan de Boer, Jos? e
Figueroa-O ’Farrill, Jelle Hartong, Akash Jain, Gerben
Oling, Niels Obers, Stefan Prohazka, Ashish Shukla, and
Amos Yarom for useful discussions. The work of J. A. is
partly supported by the Dutch Institute for Emergent
Phenomena (DIEP) cluster at the University of Amsterdam
via the programme Foundations and Applications of
Emergence (FAEME). The work of E. H. was supported
by Jelle Hartong ’s Royal Society University Research
Fellowship (renewal) “Non-Lorentzian String Theory ”
(URF\R\221038) via an enhancement award and by
Villum Foundation Experiment project VIL50317,
“Exploring the wonderland of Carrollian physics: Extreme
gravity, spacetime horizons and supersonic fluids. ”PHYSICAL REVIEW LETTERS 132, 161606 (2024)
161606-5j.armas@uva.nl
?emil.have@nbi.ku.dk
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161606-7
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