- 1Institute for Theoretical Physics and Delta Institute for Theoretical Physics, University of Amsterdam, Amsterdam, Netherlands
- 2School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh, United Kingdom
- 3Nordita, KTH Royal Institute of Technology and Stockholm University, Stockholm, Sweden
- 4The Niels Bohr Institute, Copenhagen University, Copenhagen, Denmark
- 5Science Institute, University of Iceland, Reykjavík, Iceland
- 6Institute for Theoretical Physics, Utrecht University, Utrecht, Netherlands
Carroll symmetry arises from Poincaré symmetry upon taking the limit of vanishing speed of light. We determine the constraints on the energy-momentum tensor implied by Carroll symmetry and show that for energy-momentum tensors of perfect fluid form, these imply an equation of state
1 Introduction
In cosmology, the equation of state of a perfect fluid determines the evolution of the Universe. The parameter w relating the pressure and energy density,
where we suppressed the spatial derivatives ∂iϕ as they are treated as inhomogeneous fluctuations in perturbation theory. According to the inflationary paradigm, at early times the inflaton field moves slowly and one gets w = − 1 since the time derivatives
There is a different way of saying the same thing, but it relates to the Carroll limit1 that we discuss in this paper. Using the conjugate momentum fields
In the last equation we have taken c → 0 and assumed that the potential is non-zero and stays finite in the c → 0 limit, otherwise w = + 1. The leading term in the c → 0 expansion corresponds to a perfect fluid with w = − 1 and the other terms are suppressed in terms of higher powers in c. It is important that in this limit the momentum πϕ stays finite instead of the quantity
There is an intuitive argument as to why we might want to consider small values of the speed of light. According to the Hubble-Lemaître law, the recessional velocity v of an object with respect to an observer separated by proper distance d, is given by v = Hd. If the object is far outside the Hubble sphere defined by the Hubble radius RH = cH−1, its recessional velocity satisfies v ≫ c. For a nice discussion and review of this argument, see e.g. [3]. Thus we are in an opposite limit of the Newtonian, or non-relativistic, regime. In cosmological terms, these super-Hubble scales define the regime in which Carrollian symmetry will arise, that is our claim. As we send the speed of light to zero, the Hubble radius goes to zero, so essentially the entire Universe becomes super-Hubble and hence Carrollian. The Hubble radius defines the causal patch of an observer, and as the Hubble radius goes to zero, the theory becomes ultralocal, one of the main characteristics of Carrollian physics.
As we will show in this paper, Carroll symmetry applied to an energy-momentum tensor that takes the form of a perfect fluid implies
The general philosophy we advocate is not only to understand the Carroll point itself, but to understand properties of the Universe in an expansion in c around the value c = 0. We build up spacetime in terms of small Hubble cells, with radius cH−1, the Hubble horizon, and as we expand away from c → 0, we make the Hubble cells local patches in which observers live. For small values of c, the physics is ultralocal, and lightcones close up, which is the hallmark of Carroll spacetime geometry. As c increases the Hubble radius grows, so that more and more degrees of freedom can enter the Hubble horizon and we build up the relativistic properties of the Universe.
A remark is in order on what we mean when we send the speed of light to zero, since only dimensionless parameters have physical meaning. Thus in practice we take limits in which the ratio c/vc goes to zero with vc a characteristic velocity of the setup in question2. When considering the dynamics of particles in the Carroll limit, vc is obviously the velocity of the particle, while in e.g. Carrollian electrodynamics it will be an appropriate combination of the size of the electric and magnetic fields. As explained above, in the context of cosmology we can consider the recessional velocity vc = Hd of an observer at some distance d outside the de Sitter horizon. There might also be interesting situations in condensed matter models, in which c is not the speed of light but instead a Fermi-velocity, or a sound velocity. Supersonic phenomena should then obey the constraints imposed by Carroll symmetry, we expect.
The Carroll symmetry of the limiting point where c = 0 may provide an organizing principle in the study of the perturbative expansion around it. For instance, this vantage point has recently proven to be beneficial in the study of the opposite limit, namely the non-relativistic limit c → ∞. More concretely, a non-relativistic expansion of the relativistic Lie algebra reveals the underlying structure of the 1/c2 expansion in general relativity coupled to matter [4–6]. Although we will not address exhaustively the c2 expansion in the present work, we do adopt this approach to obtain useful insights into Carroll scalars and electrodynamics. Notably, we construct a Lagrangian that reproduces the magnetic sector of Carroll electrodynamics, which was previously unknown. More generally, before turning to these field theory considerations, we briefly review as an aid to the reader Carroll symmetry and Carroll particles. This part will also include some new results. In particular, we show that a zero energy particle always moves while a nonzero energy particle cannot move in space.
The Carroll limit and corresponding symmetry algebra were initially studied in [1, 2]. It turns out that this limit is also non-trivial from the following points of view, e.g., in [7] it was demonstrated that non-trivial dynamics for coupled Carroll particles can be realized and in [8–11] Carrollian field theories were studied. Models with tachyonic aspects respecting Carrollian symmetries were considered in, e.g. [12–14], and will furthermore feature in this work. Aspects of Carrollian gravity have received attention in e.g. [15–29].
As has been argued in this introduction, in the context of cosmology, the manifestation of Carroll symmetry in hydrodynamics can be important. Carroll fluids have previously been addressed in [30–33].
The outline of this article is as follows. In Section 2 we review some basics of Carroll transformations and representations of the Carroll algebra. We furthermore investigate the Carroll limit of de Sitter space. In Section 3 we treat Carroll particles and their realisation from an extended phase space approach. We delve into Carroll field theory in Section 4 and through an expansion in small c we systematically obtain Carroll boost invariant field theoretical realisations of scalar fields and Maxwell fields. We notably present an action that yields the equations of motion for the magnetic Carroll section. In Section 5 we establish the general equation of state
1.1 Note Added
As this work was nearing its completion the preprint [34] appeared. In that work the authors consider a method to obtain inequivalent Carroll contractions of Poincaré invariant field theories at the level of the action. Our results of the Carroll ‘magnetic’ and ‘electric’ contractions of the scalar action and the Maxwell action, presented in Section 4, overlap and agree with their results. However, rather than adopting a Hamiltonian perspective, we employ a complementary approach as we view these contractions as arising in the context of an expansion around c → 0.
2 Carroll Transformations and Carroll Limits
2.1 Carroll Transformations and Algebra
It might seem counter-intuitive that the limit c → 0 gives something non-trivial, but it is well-documented in the literature (see e.g. Refs. [1, 7–9]) that such a limit can be taken on the Lorentz boosts to yield
This can easily be derived from the Lorentz boost generators
An interesting consequence is that velocities transform under Carroll boosts as rescalings,
This implies one has to consider the cases of zero and nonzero velocities separately, as they are not related by Carroll boosts, contrary to the Lorentzian and Galilean cases. A useful quantity is the unit-norm velocity vector
which has the property that it stays finite in the zero velocity limit. It is boost invariant, up to a possible sign flip that can arise if the boost parameter is large enough, i.e. for
The Carroll boosts together with the translations and spatial rotations form the Carroll algebra. The Hamiltonian in the Carroll algebra is a central charge. It commutes with all generators, and appears as a central charge in the commutator of translations and boosts:
with H = ∂t and Pi = ∂i. Observe that the Hamiltonian is Carroll boost invariant since
Because of translation symmetry in space and time, there will exist a conserved energy-momentum tensor. In the case of Lorentz symmetry, the energy-momentum tensor of a relativistic system is symmetric in the indices, and so the question arises: what is the equivalent for Carroll symmetry? The rotation symmetry of course still requires symmetry in spatial indices, but the Carroll boost will give something new. Writing the generators of the Lorentz and Carroll boosts as
whereas from imposing Carrollian symmetry we get3
Notice that (Eq. 8) follows from (Eq. 7) by taking c → 0 while keeping
It will be useful to have the Carroll-covariant transformation law on the energy-momentum tensor. Under a general coordinate transformation xμ → x′μ = x′μ(x) we have,
where we used the Carroll boost transformations (3) and one can further simplify these expressions by setting
The fact that in Carroll spacetime
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FIGURE 1. Consequences of a zero energy flux. Left: consider a particle with energy E enclosed by a volume V. If the particle can move, it could leave V and energy inside V is not conserved unless E = 0. If the particle cannot move, then there can be a non-zero rest energy E0 which stays inside V. Particle decay of a particle with non-zero rest energy can also not happen. Right: interactions are possible, but only between particles of zero and non-zero energy, or between particles with zero energy. In this figure, particle 2 is attracted to particle 1, but to be consistent with zero-energy flux through V2, it must have vanishing energy E2 = 0. When it enters V1, the total energy inside V1 is then still conserved. Particle 1 is all the time at rest with rest energy E1. Under Carroll boosts, energies stay the same, but velocities are rescaled.
2.2 Carroll Transformations From Lorentz Transformations
In this subsection, we derive the Carroll transformations from the Lorentz transformations by taking the limit of vanishing speed of light. We denote the Lorentz boost transformation parameter by
with
The parameter
These are the Carroll boosts. Under two consecutive Carroll boosts with parameters
In other words, in a Carroll world time is relative, in contrast to the Galilean case, where space is relative. If a particle travels over a distance
Since we can have Δt > 0 and Δt′ < 0, two Carroll observers do not necessarily agree on which event happened first. This sounds like a violation of causality, but it is not because for it to be a violation of causality physical information would have to be sent from one event to the other and since they are outside each other’s Carroll light cone (as a result of the physical separation and the fact that light cones are lines at a fixed point in space) this would require a particle with a nonzero velocity. In a Carroll Universe the latter is tantamount to a tachyonic particle. We will show further below that such particles exist and they in principle can lead to a violation of causality.
Spacetime distances become spatial distances in the c → 0 limit. The Minkowski metric degenerates as
Now we consider the contraction for the Lorentz transformations on energy and momentum
with
These are indeed the correct transformation rules in Carroll spacetime (as dictated by the Carroll algebra), and we have rederived that the Hamiltonian is Carroll boost invariant.
We can also look at the velocity of a particle,
Taking the limit of (Eq. 16) yields
which was also obtained directly in Carroll spacetime, see (4). As we mentioned there, the transformation law of velocity is quite strange. If the velocity was zero, it remains zero after a (finite) Carroll boost. These are the particles, considered e.g. in [7]. If it was non-zero, we can boost it to any other value, as large as we want. Hence we have two distinct classes, not related by Carroll boosts: particles with zero velocity and particles with non-zero velocity. We can even change the direction of the velocity vector under a large enough Carroll boost. That is by itself not surprising, but for small enough boost parameters with
Particles with non-zero velocity in the Carroll limit, satisfy v > c → 0, and hence they are tachyonic. As discussed above, they can cause violation of causality. Clearly this is unphysical. The strict c = 0 limit is unphysical. But this is not so different from the opposite limit, c → ∞. The strict c = ∞ limit is an unphysical theory as well, as it causes action at a distance. It nevertheless is useful to study the symmetries and dynamics that emerge in this limit. The proper way of thinking about both Carrollian and Galilei limits is to consider them as expansions in c/v and v/c respectively. We will discuss in more detail the dynamics of Carroll particles in the next section.
2.3 Representations of the Carroll Algebra
We will next discuss representations of the Carroll algebra in analogy with the massive and massless representations of the Poincaré algebra that are characterised by the eigenvalues of the momentum squared and the norm of the Pauli–Lubanski vector. Our results agree with those of [35].
The Carroll algebra consists of the generators H (Hamiltonian), Pi (spatial momenta), Ci (Carroll boosts), and Jij = − Jji (spatial rotations). The nonzero commutators are given by
where i, j, k, l run over 1, …, d. Representations are labelled by the eigenvalues of the central element H and the quartic Casimir
We will momentarily specialize to d = 3 and be quite explicit. Before doing so, we can describe the representations for general d a bit more heuristically. There are basically two cases, H = 0 and H ≠ 0. When H = 0, P and C commute and so they can be simultaneously diagonalized and states can be labeled by their eigenvalues. Rotations act on these eigenvalues and, possibly, simultaneously on an internal vector space. Once we extract an irreducible representation of the rotation algebra we can construct an irreducible representation of the Carroll algebra. In the absence of the internal vector space the zero energy irreducible representations can be labeled by PiPi, PiCi, and CiCi.
For H ≠ 0 the commutation relations of Pi and Ci are just like those of coordinates and momentum and can be represented by wave functions of Pi or equivalently Ci. We can combine these wave functions with a representation V of the rotation group, and then the rotation group acts simultaneously on this representation and by rotating Pi and Ci. States in the irreducible representation take the form |Pi⟩ ⊗ |ψ⟩ with
We now describe the case of d = 3 spatial dimensions more explicitly. For further details we refer to Supplementary Appendix SA. Here we summarise the main points.
For d = 3 it is useful to define Wk and Sk via
where ɛijk is the Levi-Civita symbol. In d = 3 we can also define the operator L = SiPi (which is closely related to the helicity operator). The Carroll algebra is a semi-direct sum of the Abelian ideal spanned by {H, Pi} and the Euclidean algebra
• When E ≠ 0 we can always go to a frame where pi = 0 by performing a Carroll boost. In this case the little group is SO(3) and the eigenvalues of WiWi are E2s(s + 1) with s = 0, 1/2, 1, …. We can always go to a frame in which the states are of the form
• When E = 0 the momentum pi is Carroll boost invariant. Using a rotation we can without loss of generality set
Further below we will see examples of either of these two representations.
2.4 Carroll Limit of de Sitter Space
In gravity, spacetime is curved, so it is important to study the Carroll limit of Lorentzian metrics. For the purposes of this paper, we look at one particular example relevant for cosmology, namely de Sitter space consisting of pure dark energy. The exponentially expanding Universe is described by the de Sitter metric, which in planar coordinates takes the form
with H the Hubble constant and with isometry group SO(4, 1) in four spacetime dimensions. The isometry group contains three rotations and three spatial translations, and furthermore the isometries corresponding to scale and special conformal transformations which infinitesimally take the form
All factors of c are made explicit here, and so in the limit c → 0, keeping H fixed, the b-isometry becomes the Carroll isometry
leaving the Carroll metric4
The metric in (Eq. 24) is written in planar coordinates and they fit into the FLRW ansatz. One can also use conformal time, τ = − 1/(aH), then we obtain the Carroll metric
which is again the metric on three-dimensional Euclidean space, up to a conformal factor.
3 Carroll Particles
In this section we consider various aspects of Carroll particles. Point particles can conveniently be described in an extended phase space system, which is useful for Carroll particles (see e.g. also [7]) as generic Carroll systems do not seem to have a well-defined initial value problem with respect to coordinate time. This follows from the fact that, as discussed in the previous section, Carroll transformations can change the direction of trajectories and even make them frozen in time. To have a well-defined initial value problem one can employ particles which use some internal clock time (cf. proper time) which can be used to define dynamics.
3.1 Extended Phase Space
We first briefly recap how the extended phase space formulation works for ordinary particles. Consider the worldline action for a particle given by
which describes an extended phase space with d + 1 coordinates and d + 1 momenta (d equals the number of spatial coordinates) and zero Hamiltonian. Notice that the combination
To reproduce some familiar systems from this we consider the class of actions
where Oα are some constraints with Lagrange multipliers λα and H is some Hamiltonian on the extended phase space. In order for this to make sense we need that {H, Oα} = cαβOβ so that time evolution preserves the constraint surface. We can have first class (commuting on the constraint surface) or second class constraints. In the first case the Lagrange multipliers will not be fixed as a reflection of the gauge symmetry which first class constraints typically generate. In the second case the Lagrange multipliers will typically be fixed by the constraints and the equations of motion, and we can pass to the reduced phase space. One way to find the actual dynamics of the system, is by examining the resulting field equations.
If the system has a set of symmetry generators on the extended phase space, the constraints and Hamiltonian must be compatible with these symmetries in order to preserve them. This can be studied canonically (i.e. through Poisson brackets of Oα and H with the symmetry generators) but also through the action (i.e. verifying that it is invariant under these symmetries). Now, let us consider some cases:
1) O1 = E − p2/2m, H = 0. This includes a first class constraint. The field equations are
and one sees that these still depend on λ1. The action is s-reparametrization invariant, where one also needs to transform λ1. We can eliminate λ1 to obtain
where the dot in the last equation now stands for the derivative with respect to t.
2) O1 = E − p2/2m, O2 = t − s; H = 0. We now have second class constraints and the field equations are
If we combine these with the constraints we find λ2 = 0 and λ1 = 1, so the Lagrange multipliers are completely fixed. Again, one finds a standard non-relativistic particle, since the extra constraint simply relates t to s but does not have any essential impact.
3) O1 = E, H = p2/2m. This includes a first class constraint plus a non-trivial Hamiltonian. The field equations are
which is once more a standard particle. However, t is no longer the usual time variable, as this role has been taken over by s. In fact the dynamics in t and E completely decouples. The energy is not given by E but by the Hamiltonian H.
4)
which is just the Lagrangian in first order form. So such a first class constraint corresponds to standard dynamics in general.
3.2 Carroll Symmetry and Single Carroll Particles
We now apply the extended phase space technique to the dynamics of Carroll particles, starting with a single particle. The standard Carroll algebra has generators
for time translations, Carroll boosts, space translations and rotations respectively. It follows again that E commutes with the other generators, and hence is a central charge. One could imagine more complicated realizations of this algebra on a given phase space, but we will restrict to this simple choice.
To add a constraint consistent with Carroll symmetry, it must Poisson commute with these generators. Because of time and space translational invariance (or Poisson commutability with E and
giving rise to the equations of motion
This corresponds to the Carroll particle at rest. It has zero velocity and constant energy E0. This is to be expected since in the Carroll limit, the light-cone closes and only particles with zero velocity can survive the c → 0 limit.
There is, however, an interesting alternative choice corresponding to the introduction of two constraints, E = 0 and
such that the action becomes
Indeed, from the Lagrangian follows the momentum
which satisfies the contraint
as it should. The equation of motion says that
We thus confirm the properties concluded in the previous section as a consequence of having a zero energy flux: If particles can move, they must have zero energy, and if they have non-zero energy, they cannot move and there is only rest energy. Having established this, we now turn to the question of how these particles can arise from a relativistic theory by taking the limit c → 0.
3.3 Carroll Particles From Relativistic Particles
Let us consider a free relativistic particle with energy E, mass m and momentum
and hold in any Lorentz frame.
We now encounter a subtlety when taking the Carroll limit. The gamma-factor associated with the particle with velocity
Notice that this is a different object compared with the gamma-factor in the Lorentz transformation that contains the boost parameter
If we keep u fixed and non-zero, this expression becomes imaginary in the Carroll limit. It seems there are two options: either we require that the Carroll velocity identically vanishes, u = 0, and then γu → 1, or we allow for imaginary values
where we allow for the two branches of the square root. This expression for γu vanishes in the strict Carroll limit, but this is the leading term in the small c expansion.
The case of zero velocity,
We now consider the more interesting case in which
This can only make sense and yield real and non-zero values of the momentum if we keep mc fixed and the original relativistic particle was a tachyon, so
for some real p0 and therefore m2 < 0. That tachyons enter into the picture is not so surprising since (45) also follows from u2 > c2. Notice furthermore that the scaling law on the mass m is different as compared to the case of the relativistic massive particle with zero velocity, where the mass m scales like ϵ−2 in the Carroll limit. The momentum of these tachyon-like particles satisfies the constraint
Notice the ± sign in the momentum, which is a consequence of the fact that the momentum as a function of the velocity is no longer single-valued (as follows from (Eq. 45)). The velocity u is undetermined and arbitrary but non-vanishing. These Carroll particles, for p0 ≠ 0, cannot be put to rest by a boost, they always move.
It is interesting to contrast Eq. 49 with (Eq. 41). The latter does not have a sign ambiguity. To go from (41–49) we need to divide the numerator and the denominator of (Eq. 41) by
A similar representation with zero energy can be found from the Carroll limit of massless particles. These have dispersion relation E = pc, and for fixed p, we have E → 0 in the Carroll limit.
3.4 Many-Particle Carroll Systems
It is relatively easy to generalize the extended phase space approach to many particles which we label by a = 1, …, N. The action on extended phase space is now
where we sum over a (the number of particles) and α (the number of constraints). To get a standard classical mechanical system of many particles, we can either choose a set of first class constraints
We now apply this to a system of many Carroll particles, as was also considered in Ref. [7]. In the case of N particles, and can realise the generators of the Carroll algebra as
Apart from rotations, the following set of building blocks commute with the Carroll algebra:
but also the combinations
with tab = ta − tb and
To construct an interacting Carroll system, we need to find a set of commuting constraints, or as shown above, a set of commuting constraints plus a Hamiltonian. In fact, we only need that the variation of the Hamiltonian and the constraints is proportional to the constraints, as in that case we can make the theory invariant by letting the Lagrange multipliers transform.
For Carroll symmetry this means that we can take first class constraints of the form Ea − H where H is now some rotationally invariant function of the invariants (52). Any H which is a function of (Eq. 52), and rotationally invariant, can in principle be used, giving rise to a fairly rich spectrum of possible interacting Carroll particles. This does not contradict the representation theory as that only applies to the center of mass degree of freedom. These examples were studied in [7].
It is also possible to take the other point of view where we for example impose first class constraints Ea = 0 for all a and add an explicit Hamiltonian H. But now something interesting happens. As long as the Hamiltonian does not involve the time coordinates, its Poisson bracket with the generator of Carroll boosts will be proportional to a linear combination of the Ea. This is not zero, but we can compensate for this by a suitable transformation rule for the Lagrange multipliers. Therefore, we need not worry about boost invariance of H, and in fact any H which preserves translations and rotations is admissible. This is no different from standard many-particle dynamics. It looks like any many-particle Hamiltonian can be dressed with a Carroll symmetric center of mass degree of freedom to provide a Carroll invariant system of particles each with zero energy. This case was not considered in [7], and it might be instructive to get these systems directly as a c → 0 limit of a relativistic many-particle system.
4 Carroll Field Theory
In this section, we switch to field theory, and prepare for the application to cosmology we have in mind. The starting point is to construct Lagrangians that are Carroll invariant and can be obtained from a c → 0 limit of a relativistic theory or through an expansion in small c. We treat here the real scalar field and the Maxwell field as examples.
4.1 Scalar Field
Consider a relativistic real scalar field ϕ. Under a Lorentz boost, it transforms as
The relativistic scalar field lagrangian density
transforms into a total derivative, as is well known. The conjugate momentum is
The relativistic energy-momentum tensor is
and is symmetric when raising the indices.
For a quadratic potential of the form
with
We now make an expansion around c = 0 so we write7
for some Δ. Defining as before
The field ϕ0 is a scalar field with respect to Carroll transformations, as the Carroll generator for boosts is Ci = xi∂t. The higher order modes in the expansion are not Carroll scalars, and transform into each other under boosts.
Using this expansion, it is rather straightforward to construct Carroll invariant Lagrangians. For free fields, we find, for the two lowest orders in the small-c expansion,
where we defined
It is easy to add interactions, starting from a Lorentz invariant potential by simply substituting the expansion 59) in the potential. Different choices can be made however, depending on how the coupling constants in the potential scale with c. We consider here the simplest example of a quadratic potential. Assuming the parameters depend homogeneously on c there are still two choices, namely keeping E0 ≡ mc2 (or ω0 = E/ℏ) fixed, or keeping the Compton wavelength λ−1 = μ ≡ mc/ℏ fixed, similar to the analysis of the relativistic particle in Section 3.3.
For fixed ω0, we get
The equation of motion associated to
For
The solutions to the response of ϕ1 to ϕ0 can be found in terms of plane waves,
Both ϕ0 and ϕ1 are fields arising in the expansion around c = 0, and we can use them to reconstruct the relativistic scalar using (59): ϕ = ϕ0 + c2ϕ1 + ⋯ , where we put Δ = 0 for simplicity. Then the full relativistic plane wave solutions are known of course, and involve the frequency in (58). We can expand the relativistic dispersion relation around c = 0 using
where we remind that ω0 = mc2/ℏ which we kept fixed in the Carroll limit. One now sees that this expansion is reproducing the expansion in ϕ using (Eq. 65), as it should. Furthermore, if we perform an infinitesimal Carroll transformation on the right hand side of the dispersion relation, we find
Now we consider the second possibility, in which we keep μ = mc/ℏ fixed in the Carroll limit. Then we find
The equations of motion for
For
If
The relativistic dispersion relation in this case is written as
To conclude this part of the discussion, we find again, just as for the relativistic particle, two representations in the Carroll limit, those with non-zero energy E0 = ℏω0 = mc2 and those with zero energy where we kept mc/ℏ fixed.
We now consider some further properties of
4.1.1 Energy-Momentum Tensor
We start with the Carroll invariant Lagrangian
The corresponding energy-momentum tensor is10,
This energy-momentum tensor obeys the Carroll transformation laws (Eq. 9) and the constraint that
where the left-hand side stands for the relativistic energy-momentum tensor which transforms as a tensor under Lorentz transformations. On the right-hand side, the leading term
The theory described by the next-to-leading order action has the Lagrangian
where we ignored possible potential terms. Using (71), adapted to a theory containing two scalar fields, the energy-momentum tensor components now read
One can check that it is conserved, but it does not satisfy the Carroll constraint
The expansion of the relativistic scalar field Lagrangian around c = 0 leads to (after appropriately rescaling the Lagrangian with c2−2Δ) the Lagrangians
with Lagrange multiplier
where the new Lagrange multiplier is χ.
We can obtain the latter Lagrangian also directly by taking a Carroll limit. Let us consider the following relativistic Lagrangian density consisting of two scalar fields χ and ϕ:
From the equation of motion for χ,
it is easy to see that the Lagrangian density in (Eq. 78) is engineered such that integrating out χ yields a canonical relativistic scalar field Lagrangian as given in (Eq. 55). And of course, χ is identical to the conjugated momentum when seen in first order formalism. It is however not a scalar field under Lorentz transformations.
In the Carroll limit c → 0 keeping both χ and ϕ fixed, we find that the Lagrangian density in (Eq. 78) becomes
The action is invariant under Carroll boosts if we assign the transformation laws
The field equation of χ sets
Notice that the Lagrangian (80) is actually very similar to (74). It seems that all one needs to do is to perform the field redefinition
It is conserved on-shell and indeed, as required by Carroll symmetry,
4.2 Maxwell Theory
We can also consider Carrollian versions of Maxwell’s theory. In analogy to the scalar field theory case treated in the previous section, we present here again two perspectives. One in which we consider an expansion for slow speed of light and the other based on taking the c → 0 limit, as has been studied in e.g. [8–11, 43]. Furthermore, in parallel with similar investigations of the non-relativistic expansion/limit (c → ∞) of Maxwell theory (see e.g. [44, 45]), one finds two distinct sectors, the electric and the magnetic.
4.2.1 Expansions
The Maxwell field Aμ transforms under Lorentz boosts as
where
for some Δ, and likewise for Ai with the same Δ.
Again we define
where the transformations for n = 0 are included using
Next we study the expansion of the Maxwell Lagrangian
where the field strengths are
Inserting the expansion of the vector field we thus find that the Lagrangian expands according to
with
The next to leading order Lagrangian is
where
One can explicitly check that
Finally, the Bianchi identity ϵαλμν∂λFμν = 0 implies
where
4.2.2 Electric Carroll Sector
Let us first focus on
which are respectively Gauss’ law and Ampère’s law with the Ampère term switched off, as is known for the electric limit of Carroll electromagnetism [8, 46]. In addition we have the Bianchi identities (95) giving
Furthermore, using the transformation rules of
which indeed leave the equations of motion of the electric Carrollian sector invariant. The corresponding energy-momentum tensor is given by
where we used the following improved formula for the energy-momentum tensor such that
The resulting energy-momentum tensor is Carroll covariant and remains traceless in 3 + 1 dimensions. This result coincides with what one would get from taking the non-relativistic limit of the Lorentzian energy-momentum tensor.
Let us consider general solutions to the electric Carroll sector. From (Eq. 96) we find
where
From the first condition in (Eq. 97) we obtain the following solution for ψi:
Let us consider again the electric Carroll equations of motion given by Eqs 96, 97, but this time in momentum space. These imply that
If we insert a plane wave profile
It follows that
4.2.3 Magnetic Carroll Sector
Next we turn to
which exhibit respectively Gauss law and Ampère’s law, but with the electric field
Let us consider the case when we force the leading term in the expanded action to be zero, i.e.
which indeed leave the equations of motion of the magnetic Carrollian sector invariant.
We can also find the action for the magnetic Carroll sector using a limiting procedure analogous to the one considered for the scalar field. For this we first introduce an altered Maxwell action containing a new field χi which when integrated out yields the original Maxwell action, i.e.
Here χi has to transform in a particular manner such that the Lagrangian remains invariant under Lorentz boosts. Now taking c → 0 we obtain Carroll boost invariant Lagrangian
Here the magnetic part of the Maxwell tensor survives the Carroll limit. Furthermore, one has to require the following transformation of χi under Carroll boost:
in order to keep the Lagrangian invariant under the Carroll boost. Note that the Lagrangian is precisely of the form of
The equations of motion are easily computed. First of all the Lagrange multiplier χi enforces Ei = ∂iAt − ∂tAi = 0 and hence ∂tFij = 0. The remaining equations of motion are then
Together with the Bianchi identity for the B-field,
we thus see that these are the correction equations of motion for magnetic Carroll upon identifying χi with the (true) electric field (
We also give here the resulting energy-momentum tensor of the novel action (Eq. 109) for magnetic Carroll. This takes the form
upon using Ei = 0 as well as the improved energy-momentum tensor formula
This energy-momentum tensor remains traceless in 3 + 1 dimensions and correctly satisfies
Another way to view both sectors is by starting from a relativistic Maxwell energy-momentum tensor and considering the following dimensionless combinations
while keeping the relativistic momentum density flux
4.3 Comments About Carrollian Limits of General Relativity
In both the scalar field and the Maxwell case we have seen that there are two types of Carroll limits. It is thus natural to wonder whether the same is true for general relativity. In [15] a Carroll limit of general relativity (in ADM variables) has been worked out leading to a theory with just kinetic terms (extrinsic curvature squared terms) as well as a cosmological constant, but thus without the spatial Ricci scalar term. The natural suggestion is that the other Carroll limit of general relativity requires a Lagrange multiplier term that sets the extrinsic curvature to zero on shell and which does contain a spatial Ricci scalar as well as a cosmological constant. We will report on this and other curved Carroll spacetime topics in [42].
5 Perfect Fluids
In the remainder of this paper we focus on applications of Carroll symmetry to cosmology and dark energy. We take a closer look at the arguments of the introduction, using the language of perfect fluids in a cosmological setting. Hence we will couple to gravity, in particular to the FLRW metric.
We start with some general remarks about perfect fluids as discussed in [30]. For perfect fluids with translation and rotation symmetry, but not necessarily boost symmetry, the energy-momentum tensor can be written as
This looks like the standard form of the energy-momentum tensor in a relativistic theory written in lab-frame coordinates, but it is more general, and holds also in the absence of any (i.e. Lorentz, Galilei or Carroll) boost symmetry. Here
Though more general, the expression for the energy-momentum tensor in (Eq. 116) can still be used for relativistic systems such as the real scalar field discussed in the introduction. In this case the energy-momentum tensor is given by (Eq. 57) in section (4.1) and we evaluate in the FLRW metric gμν = diag( − c2, a2(t)δij). It is a straightforward exercise to put it in the form (Eq. 116), so that we read off
together with
Here i, j indices are lowered or raised with the metric hij = a2δij or its inverse. Notice that
It is Lorentz invariant and appears in the usual formulation of a relativistic perfect fluid tensor
with relativistic four-velocities satisfying UμUμ = − c2. In fact,
With
where γ−2 = 1 − v2/c2 and v2 = vivjhij. In a rest frame, in which vi = 0, we have that
We can repeat this argument (for flat space) in more general terms based on (Eqs 7, 8). Combined with (Eq. 116), we immediately find
for Lorentz symmetry, whereas from imposing Carrollian symmetry we get
provided that vi ≠ 0. Notice that the Carroll case follows from the Lorentz case by taking c → 0 while keeping
We know that the zero energy flux condition tells us that
Note that, as expected, the transformation of the velocity vector coincides with the one obtained in (Eq. 4).
Using the transformation law of the momentum density
where we used the transformation law of the velocity. Since vi and bi are independent, solving this equation requires
While the vanishing of the energy flux only told us that the product of
We thus see that for any fluid velocity the equation of state of a Carroll perfect fluid is given by
The above argument relied heavily on the assumption that the momentum density is proportional to the fluid velocity, i.e.
where the momentum density
6 Friedmann Equations and Dark Energy
In the previous section we have shown that in the Carroll limit of a perfect fluid, one recovers an equation of state with
We start by coupling a relativistic perfect fluid to dynamical gravity, in our case the FLRW metric
where a possible cosmological constant has been absorbed in the pressure and energy density. Recall that the scale factor a(t) is dimensionless. Using an equation of state of the form
Note that any explicit dependence on c has dropped out, so the Carroll limit is trivial here. The solutions are well known and one must separate w = − 1 from the rest:
with two integration constants a0 and a1, and H(t) the Hubble function which is constant for w = − 1. The Hubble radius, RH(t) = cH−1 grows linear in time for any value of w ≠ − 1.
From the Friedmann equations follow the identities, valid for any w,
where we remind that the kinetic mass density ρ follows from (Eq. 121). One can see from the first equation that
The simplest case is dark energy, w = − 1, i.e. a scalar field constant in time and space, so πϕ = 0 and vi = 0, but with a nonzero, but constant potential
For fixed Λ and GN, the Hubble constant would diverge in the Carroll limit, but it is important that H stays fixed in order to maintain exponential expansion and the conformal transformations as isometry group (see (27)). Furthermore, as explained in the introduction we want to keep the potential V = Λ fixed and finite in the Carroll limit and so we need to rescale Newton’s constant such that in the Carroll limit, the Hubble constant is kept fixed,
The Hubble radius RH then goes to zero,
We have
Here we have assumed that kB and ℏ behave the same in the Carroll limit. The conclusion of this is that de Sitter space in the Carroll limit becomes conformal to
Next we consider the second example, with w ≠ − 1, say w = 1, a free scalar field with vanishing potential V = 0. What happens when we take the zero speed of light limit? We cannot expect dark energy or inflation, yet there should be a well-defined Carroll limit with
This is easily solved by
where we took into account (133), i.e.
The SI-units for ϕ (in 3 + 1 dimensions) are (kg ⋅ m)1/2s−1 = J1/2m−1/2. In the Carroll limit, there is no need to rescale GN with a factor c2 this time, as H is already finite in the limit (see (132)). Therefore, ϕ goes to a constant in the c → 0 limit, but the momentum does not, it goes like t−1. The energy density is
It goes like
7 Inflation
In the previous section, we looked at two examples with a constant value of w. Generically, however, w is time dependent, as is the case during inflation. In this section, we address what happens to inflation in the Carroll limit. As we will show, the Carroll limit enforces the limit w → − 1, so towards a de Sitter phase. We will analyze in detail the Carroll limit of chaotic inflation (with a quadratic potential), but we expect our conclusions to be more general.
Before we consider the limit, let us quickly recap the chaotic inflation scenario. Ignoring spatial derivatives, which we address in the next subsection, the scalar field equation of motion is
with
In chaotic inflation, we start with an initial condition for which the scalar field is very large. This means that H should be very large as well. One furthermore looks for solutions in which πϕ is small at early times compared to the ϕ and H terms in (Eq. 141) which are large at early times. So ϕ varies slowly in time. We also assume that
which is in the textbooks on chaotic inflation. The second Friedmann equation is also satisfied, it can be written as
Now we reconsider the equations in the Carroll limit and in the spirit of section 4.1 we make expansions14
with all coefficients time dependent. Furthermore, we keep fixed the combinations
in the limit c → 0 such that the potential
The first equation determines the leading order solution for the Hubble factor and the scalar field, and corresponds to a de Sitter solution (since ϕ0 is constant in time). The second equation perturbs the scalar field away from being constant. The solution of the inhomogeneous equation is given by
Possible solutions to the homogeneous equation are not included, as they can be absorbed in the constant ϕ0 or set to zero by appropriate boundary conditions. With these boundary conditions, one reproduces precisely the inflationary solution for the scalar field in (Eq. 143). At order c2 in the Friedmann equation, one determines H1 and we find
which leads to
Notice that this only matches the result from inflation (the second equation in (Eq. 143)) when μc ≪ H0. This condition was needed for the validity of the approximation made in inflation, but is not needed in the derivation of the Carroll expansion. Of course we can redefine H0 but the relation with ϕ0 as given by (Eq. 146) is then lost.
The momentum πϕ stays constant in the Carroll limit,
and
The above analysis can be translated in terms of the slow roll parameters. For chaotic inflation, ϵ = η, with, in SI units,
Using (Eq. 143), we can rewrite the slow roll parameter as
so that slow roll is guaranteed for all times for which the Compton wavelength is larger than the Hubble radius, λ ≫ RH(t), so for super-Hubble scales. The Carroll limit guarantees this because RH → 0 for c → 0.
8 Scalar Perturbations in de Sitter Spacetime
We now look at perturbations of the fields, and in particular focus on scalar perturbations. In this subsection, we focus on the massless case. This is standard analysis and we follow the notation of [47]. The only minor difference in the notation is that we reintroduce the speed of light c into the expressions. An important set of quantities in the perturbation analysis are the Bunch-Davies mode functions vk at given wavenumber k appearing in the plane wave expansion of the scalar perturbation around a de Sitter background. Plane waves are given by
Here τ here is conformal time, which is given by τ = − 1/(aH) in de Sitter. In the far past, τ → − ∞, modes are supposed to start to be sub-Hubble with |kcτ|≫ 1, which for de Sitter means
so the mode will at some point later in time exit the horizon and become super-Hubble,
The important observation here is that the exiting of the mode functions from the horizon is also achieved in the Carroll limit c → 0, as expected since this limit would also shrink the comoving Hubble radius. Indeed, in the Carroll limit, the second term in (Eq. 153) dominates, and we get straightaway
which now is time-independent. So in order words, the Carroll limit gives us the behaviour of the mode-functions after crossing time. Therefore, also the correlation functions freeze out at super-Horizon scales.
8.1 Massive Scalar in de Sitter
We now consider a massive scalar field in de Sitter and switch on the spatial derivatives as perturbations in the scalar field. After Fourier transforming the spatial components, the Klein-Gordon field equation becomes
with wave-vector as in
As before, we make an expansion around small values of c,
and find to leading order
which is easily solved by
The functions f± can depend on k and can be determined by normalization conditions, similar to the massless case. Observe however that now there is a small time-dependence, and the fluctuations do not freeze out in the strict Carroll limit. In the zero mass limit, they however do, as ω0 → 0 and λ+ → 0, consistent with the massless case if we choose boundary conditions that dissallow the λ− solution (which goes like e−3Ht in the massless case and diverges when t → − ∞).
At next order, the equation for ϕ1 is
which is solved by (dropping again integration constants)
For νϕ = 1, this solution does not hold and is replaced by
Grouping things together, we find for the solution for the massive scalar in the Carroll expansion (with νϕ ≠ 1)
The leading term proportional to
9 Outlook
In this paper we have explored various aspects of Carroll symmetry. The Carroll symmetry algebra arises in the c → 0 limit of the Poincaré algebra. As it stands, this limit appears to be very unphysical, as particles with finite velocities become tachyonic, the usual Lorentz factor
A second situation in which the c → 0 limit is meaningful is when the relevant “effective velocity” is related to the time dependence of classical field configurations. For these there is no issue organizing the theory as an expansion in c as we illustrated in various examples in Section 4.
An alternative perspective on the role of the c → 0 limit is that in this limit lightcones get squeezed and therefore spatially separated points become causally disconnected. In the cosmological context it is the different Hubble patches which become causally disconnected. This suggests a possible broader applicability of the Carroll limit to other situations where points are taken to be approximately causally disconnected, in particular to all situations where one attempts to define an S-matrix for localized asymptotic states. Such localized asymptotic states are assumed to exist in isolation and not interact with each other at early and late times, and may only exists approximately in actual quantum field theories, for example due to IR divergences. In such cases one can try to factorize the S-matrix in a hard and soft part and we speculate that Carroll-like limits may in general be of relevance to the hard part of such amplitudes, but that the E = 0 Carroll particles could be of relevance to the soft modes. We leave a further investigation of this issue to future work.
The problems that arise in taking the c → 0 limit at the level of individual point particles also show up when trying to interpret energy-momentum tensors that are compatible with the Carroll symmetries as describing the thermodynamics of a well-defined underlying quantum system. The Carroll algebra does not allow one to write a non-trivial dispersion relation which relates the energy to the momentum, and this leads to a divergence in the integral over momenta in the finite temperature partition function.15 One can try to avoid this conclusion in several ways, e.g. by restricting the integral over momenta by hand, or looking at systems with no momenta such as spacetime-filling branes, but none of these lead to a particularly compelling picture. One could also choose to take the Carroll limit directly at the level of the partition function of a relativistic gas, but this leads to a vanishing result unless one adds additional factors of c by hand. In such a case one gets a finite answer which defies a direct quantum mechanical understanding unless one is willing to consider e.g. a gas of tachyons, but we do not think that this is a particularly interesting direction to explore for obvious reasons. We refer the reader to our forthcoming work [42] in which we elaborate on several of these points.
Another observation that we would like to highlight are the two types of Carroll limits which generically seem to exist. These two different limits are already visible in the Carroll algebra where the representation theory is quite different depending on whether the energy vanishes or not. These two qualitatively different behaviours also appear if we look at correlation functions. Consider for example the two-point function of two Carroll scalar fields. It is easy to see that the following two answers are both solutions to the Carroll Ward identities
where the first case corresponds to vanishing energy, and the second one to non-vanishing energy. Equivalently, the first case is one where we put the canonical momentum of a field equal to zero, whereas the second case is relevant for theories where we drop spatial derivatives, in line with the field theory limits considered in Section 4.
To quote Lewis Carroll from Alice in Wonderland: “Begin at the beginning,” the King said, very gravely, “and go on till you come to the end: then stop.” We tried to follow the advice of the King quite closely in this paper and hope to have convinced the reader that we have not quite come to the end (yet).
Data Availability Statement
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.
Author Contributions
All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.
Funding
JB is supported by the European Research Council under the European Unions Seventh Framework Programme (FP7/2007-2013), ERC Grant agreement ADG 834878. JH is supported by the Royal Society University Research Fellowship ``Non-Lorentzian Geometry in Holography'' (Grant Number UF160197). NO is supported in part by the project ``Towards a deeper understanding of black holes with non-relativistic holography'' of the Independent Research Fund Denmark (grant number DFF-6108-00340) and by the Villum Foundation Experiment project 00023086. WS is supported by the Icelandic Research Fund (IRF) via a Personal Postdoctoral Fellowship Grant (185371-051).
Conflict of Interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s Note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
Acknowledgments
We thank Eric Bergshoeff, Joaquim Gomis, Gerben Oling and Guilherme L. Pimentel for useful discussions.
Supplementary Material
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2022.810405/full#supplementary-material
Footnotes
1The c → 0 Carroll limit of the Poincaré group, being the opposite of the c → ∞ Galilean (non-relativistic) limit, was first considered in [1, 2].
2This is not the same as an ultra-relativistic limit, where c/vc → 1. The Carroll limit, which we consider in a given spacetime dimension, is therefore not an ultra-relativistic limit, a terminology that is sometimes inappropriately used in the literature.
3This is an on-shell constraint, i.e.
4More precisely, this is the spatial metric hμν of Carrollian geometry. The latter includes in addition the timelike vector (or inverse vielbein) vμ, satisfying vμhμν = 0. See e.g. [18, 36]. Correspondingly, an isometry ξμ of a Carrollian geometry is defined by the conditions
5This Carrollian spacetime is a homogeneous spacetime, referred to as the light cone in [37].
6The basic Poisson brackets on extended phase space are {t, E} = − 1 and
7Odd powers of c do not play a role in all examples we consider.
8In principle, one could generate an algebra expansion by writing
9This is a special case of a general result that states that the equations of motion of a Lagrangian at order N, say, contain all the equations of motion of the Lagrangians at orders n < N via the dependence of the Nth order Lagrangian on the subleading fields, see [5] (section 2.5).
10We compute the energy-momentum tensor from the Noether procedure,
11The next-to-leading order Lagrangian is invariant under a symmetry group whose Lie algebra can be obtained by expanding the Poincaré algebra around c2 = 0 and quotienting this algebra by keeping only the level zero and level one generators (see [42] for more details).
12This is also natural when considering the 1-form A = Aμdxμ = A0dx0 + Aidxi = Atdt + Aidxi, while it also ensures we can write
13By vanishing Hubble radius, we mean it is smaller than any other length scale in the problem. In empty de Sitter, there is however no other length scale, so we rescale RH → ϵRH and send ϵ → 0. We remark furthermore that our analysis is classical. In a quantum theory, one could compare with the Planck length. Then the classical Carroll regime is valid for small Hubble radius, but still larger than the Planck length.
14One could, as in section 4.1, introduce additional overall scaling factors
15This same conclusion would apply to massless Galilean particles as well, since in that case only the Hamiltonian would transform under Galilean boost, and not the momentum. It is in fact the mass, or availability of a central extension, in the Bargmann case which enables momentum to transform as well under Galilean boost. In general dimensions, a central extension like this is unavailable for Carroll, see e.g. [7].
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Keywords: carroll symmetry, inflation, dark energy, field theory, carrollian dynamics
Citation: de Boer J, Hartong J, Obers NA, Sybesma W and Vandoren S (2022) Carroll Symmetry, Dark Energy and Inflation. Front. Phys. 10:810405. doi: 10.3389/fphy.2022.810405
Received: 06 November 2021; Accepted: 20 January 2022;
Published: 08 April 2022.
Edited by:
Dieter Van Den Bleeken, Boğaziçi University, TurkeyReviewed by:
Arjun Bagchi, Indian Institute of Technology Kanpur, IndiaMario Trigiante, Politecnico di Torino, Italy
Copyright © 2022 de Boer, Hartong, Obers, Sybesma and Vandoren. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Niels A. Obers, obers@nbi.ku.dk