Massive carrollian fields at timelike infinity

Motivated by flat space holography, we demonstrate that massive spin-$s$ fields in Minkowski space near timelike infinity are massive carrollian fields on the carrollian counterpart of anti-de Sitter space called $\mathsf{Ti}$. Its isometries form the Poincar\'e group, and we construct the carrollian spin-$s$ fields using the method of induced representations. We provide a dictionary between massive carrollian fields on $\mathsf{Ti}$ and massive fields in Minkowski space, as well as to fields in the conformal primary basis used in celestial holography. We show that the symmetries of the carrollian structure naturally account for the BMS charges underlying the soft graviton theorem. Finally, we initiate a discussion of the correspondence between massive scattering amplitudes and carrollian correlation functions on $\mathsf{Ti}$, and introduce physical definitions of detector operators using a suitable notion of conserved carrollian energy-momentum tensor.


Introduction
More than two decades after the advent of the AdS/CFT correspondence [1][2][3], substantial effort is currently devoted to the investigation of the holographic principle in more general settings.In particular the program of celestial holography aims at providing a holographic description of quantum gravity in asymptotically flat spacetimes in terms of a somewhat exotic conformal field theory (CFT) defined on the two-dimensional celestial sphere CS 2 [4][5][6].Within this perspective the Lorentz group is realised as the group of global conformal transformations of the celestial sphere CS 2 , while translations are realised as internal symmetries connecting conformal primaries of different scaling dimensions.Alternatively one can choose to realise the full BMS group [7][8][9] and its Poincaré subgroup as conformal isometries of the spacetime null conformal boundary I ∼ = R × CS2 , yielding a variant of flat holography called carrollian holography.This has allowed in particular to interpret massless scattering amplitudes as a set of correlators of a three-dimensional conformal carrollian theory living on I [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24].Although this carrollian approach has attractive features, only massless particles have been accounted for thus far, and the purpose of the present work is to provide a carrollian description of massive particles.
Originally obtained as the vanishing speed-of-light limit of the Poincaré group [25,26], carrollian symmetries have since played a role in the description of a wide array of physical phenomena, ranging from black hole physics [27][28][29][30] to exotic condensed matter systems [31][32][33][34][35]. Concretely, the Carroll algebra may be obtained by contracting the Poincaré algebra by reinstating factors of the speed of light c in the Minkowski metric that appears in the commutation relations, and then taking the limit c → 0 (see Fig. 1).The resulting algebra has the same number of generators as the original Poincaré algebra.In (d + 1) spacetime dimensions the nonzero brackets of the Carroll algebra are [J ab , J cd ] = δ bc J ad − δ ac J bd − δ bd J ac + δ ad J bc , [J ab , P c ] = δ bc P a − δ ac P b , where a = 1, . . ., d is a spatial index, and where (H, P a ) generate spacetime translations, C a are the generators of carrollian boosts and J ab are the generators of d-dimensional spatial rotations.In the same way that a lorentzian geometry locally realises Poincaré symmetry, a carrollian geometry locally realises Carroll symmetry. 1In a carrollian geometry, the notion of a metric gets replaced by a carrollian structure, which consists of a vector field and a degenerate spatial "metric".
Lightcones opening and closing as c → ∞ and c → 0, respectively.In the galilean limit c → ∞, the lightcone opens up and Einstein's principle of relativity gives way to the well-known galilean relativity principle.In the carrollian limit c → 0 on the other hand, the lightcone closes up, implying that physical entities only move along the time direction.In the intermediate regimes where c ≪ 1 and c ≫ 1, physical systems are described by expansions in either c or 1/c (see, e.g., [39][40][41][42]).
A fundamental ingredient of holography is that states in the Hilbert space of the theory can be carried both by fields living in the spacetime bulk and by conformal fields living at the spacetime conformal boundary.Indeed, massless particle states can be encoded into massless fields in (d + 1)-dimensional Minkowski space M d+1 or into carrollian conformal fields on 1 , a simple fact underlying the very foundation of carrollian holography [11,15,19].However, such carrollian conformal fields cannot possibly carry massive particle states as already pointed out in [19], since the latter propagate to/from timelike infinity i ± rather than null infinity I .Hence it is natural to look for a holographic description of massive particles at timelike infinity, or more precisely on the geometry which appropriately captures the relevant structure near timelike infinity.The two points i ± in the conformal compactification of Penrose obviously do not provide enough structure to properly describe massive particle states.We will argue that the appropriate geometric structure is that of Ti d+1 ∼ = R × H d [43], or equivalently carrollian anti-de Sitter [44][45][46], which is a curved carrollian manifold of the same dimensionality as the spacetime bulk M d+1 itself.The space Ti d+1 is the blowup of i ± in perfect analogy with the Ashtekar-Hansen structure used for spatial infinity i 0 [43,[47][48][49].The geometry of Ti d+1 can be viewed as a fibration over the hyperbolic space H d , which will end up playing the role of the momentum mass shell for massive particles.While H d admits a natural action of the Lorentz group SO(d, 1) as its isometry group, the additional null direction along the fibres allows for a nontrivial representation of the abelian translation subgroup T d+1 as well.In this work we will develop an intrinsic description of Ti d+1 and the associated carrollian fields encoding massive particle states, and subsequently demonstrate that these indeed arise when considering massive fields in Minkowski spacetime in their limit to timelike infinity.This will provide an "extrapolate dictionary", by which we mean that carrollian fields on Ti d+1 are obtained by taking the late time asymptotic limit of massive fields in M d+1 .Conversely we will also provide an algorithm to reconstruct the massive fields in M d+1 from the corresponding carrollian fields on Ti d+1 .
Let us note that the notion of holography is usually reserved for a description of spacetime physics through a lower-dimensional theory.At first sight what we describe in this work is therefore not strictly holographic in this sense.But even though there is no dimensional reduction in going from M d+1 to Ti d+1 , the dependence of the massive degrees of freedom on the "time" fibre R over H d is essentially trivial as will be made explicit in this work.More specifically, the time-dependence of the carrollian fields is simply fixed by the corresponding mass.The true degrees of freedom are obtained by factoring out this time-dependence, and will be shown to exactly correspond to the particle degrees of freedom on their momentum mass-shell identified with the base space H d .From that perspective what we describe is holographic, in that the true degrees of freedom live on a codimension one surface H d located in some asymptotic region of M d+1 .Another way to put it is that the momentum mass-shell H d of the massive particles can be embedded into Ti d+1 , which itself captures the physics happening in the asymptotic future of M d+1 .Scattering amplitudes are defined on the momentum mass-shell H d , which is codimension-one with respect to M d+1 , and can therefore be viewed as the local observables of a dual theory.
Given that we are interested in giving a carrollian field theory description of mas- sive particle states, i.e., unitary irreducible representations (UIRs) of the Poincaré group ISO(d, 1) [50], the latter will play a central role in our discussion.In particular all three geometries of relevance to carrollian holography, namely M d+1 , I d and Ti d+1 , can be introduced as homogeneous spaces of ISO(d, 1) [43].They can be thought of describing different parts of the Penrose diagram of Minkowski spacetime as depicted in Figure 2, and further all arise as Poincaré orbits in a higher-dimensional embedding space R d+1,2 .With such homogeneous spaces at hand, we are confronted with the task of constructing covariant field representations of ISO(d, 1) encoding the particle UIRs of interest.This can be achieved using the method of induced representations, inducing from a representation of the isotropy group of the homogeneous space, and subsequently determining the intertwining relation with one of the particle UIRs.The construction of carrollian conformal fields on I d carrying massless particle states of arbitrary spin was performed in this way in [19].In this work we will complete the story with the construction of carrollian fields on Ti d+1 encoding massive particle states of arbitrary spin.
While bulk fields encode particle states in an intricate fashion, it is a generic phenomenon that boundary fields encode them in a much more economical way.For instance the encoding of spinning massless particles into fields in M d+1 requires the introduction of a wave equation, transversality conditions and the concept of gauge redundancies, while the carrollian conformal fields on I d essentially are the massless particle states up to a single integral transform in energy [11,15,19].The story we will present for massive particles is very similar in that respect.After factorising the dependence on the carrollian time direction in Ti d+1 , which will be fully fixed by the mass of the corresponding particle, the leftover carrollian field exactly corresponds to the particle wavefunction on its momentum mass shell H d .The extrapolate dictionary alluded to above can therefore be summarised in physical terms as follows: the momentum wavefunctions of the asymptotic particle states are directly obtained by taking the limit to i ± of the massive fields in M d+1 .
In a nutshell the carrollian field theory on Ti d+1 very naturally describes the physics of asymptotic massive particles.This will be exemplified with the soft graviton theorem [51] and its formulation as the Ward identity associated with BMS supertranslations [52][53][54].In practice this formulation involves the identification of canonical charges generating the supertranslations of the various particles involved in the gravitational scattering process of interest.We will show that the charge which generates the supertranslation of the scattered massive particles [54] is a standard Noether charge from the perspective of Ti d+1 .Indeed the algebra of vector fields preserving the carrollian structure on Ti d+1 is actually much bigger than the Poincaré algebra, and in particular contains symmetries that may be identified with BMS supertranslations.Hence a Noether current is easily constructed by contracting a suitable notion of a carrollian energy-momentum tensor on Ti d+1 with a supertranslation Killing vector field on Ti d+1 , and the corresponding conserved charge will be shown to exactly coincide with the canonical generators considered in [54] in relation to the subleading soft graviton theorem.Part of the analysis will consist in giving a full characterisation of what is meant by a carrollian energy-momentum tensor on Ti d+1 .

Outline of the paper
We now give a brief outline of the paper, illustrating some of the key results by using massive scalar particles and fields.
In Section 2 we introduce the carrollian geometry of timelike infinity.In particular, after we provide a short introduction to carrollian geometry (Section 2.1), we describe how Ti = AdSC arises as a homogeneous space of the Poincaré group (Section 2.2) with carrollian invariants (Section 2.4).In Section 2.3 we also provide a useful embedding space picture.
Using these symmetries we construct in Section 3 massive carrollian fields of generic integer spin living on Ti d+1 , inducing from an irreducible representation of the isotropy group of Ti d+1 .We explain that they can be understood as reducible unitary representations of the Poincaré group and we show that imposing irreducibility by fixing the quadratic Casimir directly reproduces Wigner's irreducible representations.
In Section 4 we connect these intrinsically defined carrollian fields to the late time limit of massive fields in Minkowski space (Section 4.1).Considering a hyperbolic foliation of Minkowski space such that η = −dτ + τ 2 h H with h H the metric of a d-dimensional hyperboloid H d , we identify (∂ τ , h H ) as the appropriate carrollian structure as τ → ∞.We then provide an extrapolate dictionary relating massive fields ϕ in M d+1 to carrollian fields φ in Ti d+1 , which in the spinless case simply reads φ(τ, x) In particular we note that the second equation, inferred from the Klein-Gordon equation, amounts to an irreducibility condition as it fixes the quadratic Casimir operator acting on the carrollian field representation φ on Ti d+1 .As a result the carrollian field can be decomposed as φ(τ, In Section 4.2 we use a stationary phase approximation on the plane wave expansion of the minkowskian field ϕ to demonstrate that the carrollian modes ϕ ± are simply the momentum particle operators, where the implicit relation k(x) identifies the asymptotic hyperboloid H d to be the momentum mass shell of the corresponding particles.In Section 4.3 we connect the carrollian fields to the celestial conformal primary basis [55], providing us with a triangular relation between massive momentum, carrollian and celestial states, cf. Figure 4.
In Section 5 we provide an intrinsic definition of a carrollian energy-momentum tensor (EMT) on Ti d+1 (Section 5.1), which is then applied to the massive or "electric" Carroll scalar (Section 5.2).For the latter we also show that the intrinsic description agrees with the late-time limit of massive fields in Minkowski space and we also derive it intrinsically from the action of the electric scalar.In Section 5.3 we discuss the relation between the Ti-supertranslations and BMS symmetries in connection to soft graviton theorems.In particular we show that the charges generating BMS transformations of massive particle states are a special case of Noether charges on Ti, for an appropriate choice of Killing vector field of the form ξ = S(x)∂ τ on Ti, and with T µ ν the conserved carrollian EMT.Hence an obvious advantage of the Ti framework is that BMS symmetries are naturally incorporated as part of the kinematic symmetries, just like it happens at null infinity I .
In Section 6 we discuss further avenues of investigation.In particular we argue that scattering amplitudes of massive particles can be identified with carrollian correlation functions on Ti since they satisfy the required Ward identities.We also introduce natural definitions of detector operators measuring, for instance, the total energy density carried by scattered massive particles along any given direction from the origin of the detector, and which constitute the massive counterparts of "energy-flow" operators usually defined at null infinity I .We also comment on the inclusion of superrotations and the generalisation to curved space.

The carrollian geometry of timelike infinity
In this section, we provide a brief review of carrollian geometry.We then recall the description of the blowup of timelike infinity as a homogeneous carrollian geometry.First we review its construction as a homogeneous space of the Poincaré group and contrast it with Minkowski space, followed by a description using the embedding space formalism.

Carrollian geometry in a nutshell
A carrollian geometry is an example of a "non-lorentzian geometry", where the familiar notion of a metric is replaced with something else: in this case, the metric is replaced with a carrollian structure that realises local Carroll symmetry (1.1), instead of the local Poincaré symmetry that characterises lorentzian geometries.
There are many ways to arrive at carrollian geometry.For example, one may obtain a carrollian geometry by formally expanding a lorentzian geometry in powers of the speed of light [40,56].Another approach involves "gauging" the Carroll algebra (1.1), which from a mathematical perspective corresponds to the construction of a Cartan geometry modelled on flat carrollian spacetime considered as a homogeneous space of the Carroll group [57,58].
Regardless of its origin, a carrollian geometry is a (d + 1)-dimensional manifold M equipped with a carrollian structure consisting of a nowhere-vanishing vector field n = n µ ∂ µ and a symmetric tensor q = q µν dx µ dx ν , where {x µ } µ=0,...,d is a chart on M. The carrollian vector field n is in the kernel of the "spatial metric" q, namely n µ q µν = 0 . ( It is useful to introduce "inverses" to the carrollian structure given by τ = τ µ dx µ and γ = γ µν ∂ µ ∂ ν , which satisfy the relations In the same way that a pseudo-riemannian (or lorentzian) geometry locally realises Poincaré symmetry, a carrollian geometry locally realises Carroll symmetry.Specifically, just like the vielbeins that make up the metric transform under local Lorentz transformations, the "inverses" τ and γ transform infinitesimally under local Carroll boosts as where λ µ is the parameter of an infinitesimal Carroll boost satisfying λ µ n µ = 0. Another important geometric ingredient is the adapted affine connection ∇ that defines the covariant derivative.This covariant derivative is compatible with the carrollian structure in the sense that In general, ∇ has nonzero torsion and, in contrast to connections in lorentzian geometries, it is not Carroll boost invariant unless the "carrollian spin connection" is included as an independent degree of freedom (see, e.g., [56][57][58][59]).However, when the carrollian geometry is torsion-free, the (boost invariant) affine connection is determined in terms of the carrollian structure [57,60] Γ where Ξ µν = Ξ νµ is an arbitrary symmetric tensor satisfying n µ Ξ µν .Parenthetically, we remark that when a carrollian geometry is obtained by expanding a lorentzian geometry in powers of the speed of light, the resulting connection constructed from the expansion of the Levi-Civita connection is not boost invariant, though it is compatible with the carrollian structure (2.4) [40].Finally, for later use we introduce the natural notion of a carrollian Killing vector ξ µ as that which preserves the carrollian structure, i.e., where £ denotes the Lie derivative.

Two homogeneous spaces of the Poincaré group: M and Ti
It was shown in [43] that the blowup of timelike infinity Ti in the sense of Ashtekar-Hansen [47] is described in terms of carrollian anti-de Sitter space AdSC [44][45][46].This space arises, among other realisations, as a homogeneous space of the Poincaré group, which in (d + 1) dimensions is ISO(d, 1) ∼ = SO(d, 1) ⋉ R d+1 , where SO(d, 1) is the Lorentz group and R d+1 denotes the spacetime translations.In contrast to the construction of (d + 1)-dimensional Minkowski space M d+1 as homogeneous space, where the stabiliser is the Lorentz group SO(d, 1), the stabiliser of Ti d+1 ∼ = AdSC d+1 is ISO(d) consisting of the rotations and the spatial translations, i.e., Infinitesimally, these homogeneous spaces are captured by a Klein pair (g, h), where g is the Poincaré Lie algebra and h is the Lie algebra of the stabiliser subgroup.In a basis consisting of Lorentz transformations so(d, 1) = ⟨J µν ⟩ and spacetime translations t = ⟨P µ ⟩, where µ, ν = 0, . . ., d, the nonzero brackets of the Poincaré algebra g are Note that t forms an ideal of g since [g, t] ⊂ t.It is useful to decompose the generators of the Poincaré algebra by splitting the indices into temporal and spatial directions µ = (0, a), where a = 1, . . ., d, and writing B a = J 0a and H = P 0 .In terms of these generators, the Poincaré algebra (2.8) becomes (2.9) Comparing this with the Carroll algebra (1.1), we observe that upon excluding the last two brackets on the right-hand side, and identifying B a = P a , P a = C a and H = −H, these algebras become identical (see also [43, §4.3]).The last two brackets on the right-hand side in terms of the "Carroll generators" become which are identical in form to the additional brackets that distinguishes the isometry algebra of anti-de Sitter space so(d, 2) from the Poincaré algebra iso(d, 1).In other words, adding these two brackets to the Carroll algebra is analogous to the procedure of adding a negative cosmological constant to the Poincaré algebra, and therefore the algebra in (2.9) can be called the carrollian anti-de Sitter algebra.We remark in passing that the AdSC algebra also may be obtained from c → 0 contraction of so(d, 2); for additional details, we refer to [44].Thus, the Klein pair of M d+1 is (g, h M ), where h M = ⟨J ab , B a ⟩, while the Klein pair of Ti d+1 is (g, h Ti ), where h Ti = ⟨J ab , P a ⟩.Since the stabiliser subalgebra h in either case does not contain any nontrivial ideal, the homogeneous spaces are both effective.This means that all Poincaré transformations act non-trivially on the respective spacetimes. 2urthermore, since both Klein pairs (g, h) can be written as g = h ⊕ m with [h, m] ⊂ m, both spaces are reductive.
At this stage, one may wonder what makes the space Ti d+1 carrollian.There is a simple way to derive the invariant structure of a homogeneous space from its algebraic description in terms of the Klein pair (see for example [44]): if M ∼ = G/H is a homogeneous space of G with reductive Klein pair (g, h), the h-invariant tensors on m are in one-to-one correspondence with G-invariant tensors on M .Concretely for Ti d+1 , this means that we should look for h-invariant tensors on the vector spaces m, m * , where m * = span(η, β a ) is the dual of m, with η(H) = 1 and β a (B b ) = δ a b .The stabiliser h acts on m via the adjoint, i.e., the bracket, while it acts on the dual m * via the coadjoint action, which is defined in the following manner: if α ∈ m * and X ∈ h, then X • α = −α • ad X .Using this, we find that the brackets (2.9) lead to two h-invariant tensors of low rank on m, namely H ∈ m, which corresponds to the G-invariant vector n on Ti d+1 , and δ ab β a β b in the symmetric product of m * , which corresponds to the G-invariant covariant tensor q on Ti d+1 : precisely a carrollian structure as defined in Section 2.1.We will discuss the carrollian structure on Ti d+1 in more detail in Section (2.4).

Embedding Ti into pseudo-euclidean spacetime
The blowup of timelike infinity Ti d+1 can be embedded into pseudo-euclidean space R d+1,2 : the embedding space [43] (see also Appendix B of [18] and [61]).Indeed, Minkowski space itself, along with a litany of other spaces such as the blowup of spatial infinity Spi, null infinity I , and even the Schwarzschild solution, can be realised as embeddings in R d+1,2 [43,[62][63][64].It also plays an important role in the AdS/CFT correspondence (see for example [65,66]) since, as we shall see below, AdS d+2 may also be embedded into R d+1,2 .
In the embedding space R d+1,2 , two copies of Ti arise as the intersection of Q −ρ 2 ∼ = AdS d+2 and the null hypersurface N 0 defined by Y − = 0.
Following [43], we define the (d + 2)-dimensional space Q Λ with Λ ∈ R as the quadric given by preserves these quadrics and acts on the embedding space via the vector fields which split under (2.11) as where we notice that the vector fields in (2.14a) generate the (d + 1)-dimensional Poincaré algebra iso(d, 1).The quadric Q Λ<0 gives a copy of AdS d+2 , while Q Λ>0 describes "pseudode Sitter space", which is de Sitter space with signature (d, 2).The null quadric Q 0 can be thought of as the (generalised) lightcone of the embedding space.Next, we define N σ for σ ∈ R as the null surface Y − = σ, which is preserved by the Poincaré group corresponding to the algebra spanned by the first line above for generic values of σ.We also define the intersection which is preserved by the Poincaré group that preserves N σ .As depicted in Figure 3, the blowup of timelike infinity Ti d+1 is then captured by M −ρ 2 ,0 , where we set Λ = −ρ 2 for some ρ ∈ R for definiteness.More precisely, M −ρ 2 ,0 breaks into two orbits under the action of the Poincaré group discussed above where . (2.17) This explicitly exhibits Ti d+1 as a trivial line bundle R × H d over the d-dimensional hyperboloid H d defined by η µν y µ y ν = −ρ 2 , while the coordinate along the fibre is Y + which is to be thought of as carrollian time.For more details about this construction, we refer to [43].
Setting ρ = 1, we now introduce coordinates x a = x via where the x a parameterise the hyperboloid H d , while τ is now the carrollian time coordinate.Let us also introduce xa = x a /r which is the unit vector pointing in the direction of x a .We will use both sets of coordinates (τ, y) and (τ, x) in the remainder of this paper.
In terms of the x coordinates, the Poincaré vector fields that preserve the embedding (2.17) take the form ) ) ) where we used that x a ∂ ∂x a = r ∂ ∂r .
As we shall see in what follows, these Poincaré vector fields are a subset of the vector fields that preserve the carrollian structure on Ti d+1 .

The carrollian structure on Ti
Having constructed the carrollian space Ti d+1 both as a homogeneous space of the Poincaré group and by embedding it in R d+1,2 , we now turn our attention to the carrollian structure on Ti d+1 .Using the methods of [46], one may extract the carrollian structure from the Maurer-Cartan form associated with Ti d+1 .It is, however, simpler to derive the form of the carrollian structure (n, q) directly from the coordinates introduced in (2.18): from (2.17), we see how Ti d+1 ∼ = R × H d has the structure of a trivial fibre bundle, where the coordinate along the fibre is the carrollian time, which implies that n = ∂ τ .Similarly, the carrollian "ruler" q restricted to an equal-time hypersurface, i.e., a copy of H d , is the euclidean metric on H d .Hence, the carrollian structure (n, q) on Ti d+1 is where h ab is the metric on H d and g S d−1 is the round metric on the unit sphere S d−1 .The carrollian vector n field is in the kernel of q, i.e., it satisfies the relation (2.1).Alternatively, in the y-coordinates introduced above the hyperbolic metric reads To be able to integrate, we need the invariant volume form ε, where ε (d) is the volume form on the hyperboloid H d .Furthermore, as shown in [44], Ti d+1 ∼ = AdSC d+1 is torsion-free, and hence we find that the torsion-free affine connection (2.4) on Ti d+1 has components while Γ a bc is the Levi-Civita connection on the hyperboloid H d and Γ τ ab is left undetermined, corresponding to the arbitrary spatial tensor Ξ µν that appears in the general expression (2.4).
We may now ask what are the symmetries of the carrollian structure on Ti d+1 ; in other words, what is the set of vector fields ξ µ that satisfy the carrollian analogue of the Killing equation (2.6)?This was worked out in [46], and we recall the procedure below.Writing implying that ξ µ = ξ µ (x).The preservation of q requires that £ ξ q = 0, i.e., but since ξ µ does not depend on τ and q has no nonzero τ -components, the condition above reduces to £ ⃗ ξ h ab = 0, i.e., ⃗ ξ := ξ a ∂ a must be a Killing vector of the metric h ab on H d .The vector fields ⃗ ξ, as is well known, generate the Lorentz algebra so(d, 1).Therefore, the symmetries of the carrollian structure on Ti d+1 consists of the vector fields ξ X for X ∈ so(d, 1) in (2.19a)-(2.19b),as well as an infinite set of vector fields of the form Hence, the algebra kv(Ti d+1 ) of the vector fields that are symmetries of the carrollian structure on Ti d+1 has the form where the action of so(d, 1) on C ∞ (H d ) is the standard action of vector fields on functions, [X, S] = ξ X S, where X ∈ so(d, 1) and f ∈ C ∞ (H d ).The embedding of the Poincaré algebra iso(d, 1) into the symmetry algebra of Ti d+1 is, as explained above, achieved by choosing the function S to be of the form (2.19c)- (2.19d).These symmetries bear a striking resemblance to the BMS symmetries, which arise as the asymptotic symmetries of flat spacetime [7,8], and which play an important role in celestial holography (see, e.g., [6] for a review emphasising this perspective).The algebra of BMS symmetries of (d + 1)-dimensional asymptotically flat spacetime has the form i.e., the BMS symmetries are a subset of the symmetries of the carrollian structure on Ti d+1 .We will have more to say about this in Section 5.3, where we discuss the relation to Weinberg's soft graviton theorem.
The carrollian symmetries of Ti can be further restricted, e.g., by adding the connection as an additional structure that needs to be preserved.On the other hand they can also be enlarged, similar to generalised BMS [67], by only imposing invariance of the vector field and the volume form (2.22) or a conformal generalisation thereof (see, e.g., [68] for a discussion of several such modifications).

Massive representations and carrollian fields at timelike infinity
In this section we induce two unitary representations of the Poincaré group for generic spin and in generic dimension.The first are interpreted as carrollian fields on Ti which have to satisfy, analogous to the Klein-Gordon field in Minkowski space, a carrollian wave equation (∂ 2 τ + m 2 )ϕ a 1 ...as (x) = 0. We then show that this reducible representations can be seen as a direct integral representations of Wigner's unitary irreducible representation of massive particles [50].

Carrollian fields on Ti
We proceed with the construction of fields living on Ti d+1 .This representation will be induced from the isotropy subgroup generated by iso(d) = ⟨P a , J ab ⟩ and we will do this infinitesimally similarly to [19,69].This means that the starting point is a finite-component unitary representation ϕ a 1 ...as (0) of iso(d), where (Σ ab ) b 1 ...bs a 1 ...as acts in the corresponding tensor representation of so(d).Although much of the discussion in this subsection will apply to any tensor representation of so(d), including those with mixed-symmetry, and even to arbitrary representations of spin(d), we will focus on real totally symmetric and traceless tensor representations of integer spin.In the case of the vector fields ϕ a 1 (0) with spin 1 for instance, we have In the following, we will suppress tensor indices on ϕ and Σ.
We choose coordinates using U : Ti → G and from which we obtain a representation of the full Poincaré group by translating the field to generic points on Ti d+1 We begin by collecting the following useful relations where r 2 = δ ab x a x b and xa = x a /r is a unit vector pointing in the direction of x a .To derive the last equation we used the derivative of the exponential map where and therefore the third line of (3.4) implies that By the same logic, we have [U J ab U −1 , ϕ(x)] = Σ ab ϕ(x), which leads to On the other hand, definition (3.3) allows us to write and similarly for the spatial derivatives We can now solve this system for the unknown brackets between the generators and ϕ(x).Contracting (3.8) with xa and combining the result with (3.10) yields Next contracting (3.11) and (3.9) with xa and plugging into (3.11) and (3.9), we find In summary, the infinitesimal action of the Poincaré group on the Ti field ϕ(x) is where the vector fields ξ X for X ∈ iso(d, 1) are explicitly given in (2.19).The action of the Casimir C 2 = P 2 = −H 2 + P a P a on the Ti fields is then given by The Casimir equation C 2 = m 2 defining massive representations3 then reduces to the simple differential equation This equation of motion, which is true for any spin, ensures that our fields on Ti d+1 describe massive Poincaré particles.It is the Ti analogue of the relativistic equation of motion satisfied by massive fields in Minkowski space such as the Klein-Gordon equation (see (4.13) below).This construction is analogous to massive carrollian fields [35], which can carry any spin, with the main difference that they are valued on a flat carrollian manifold, which can be seen as the flat limit of Ti.Equation (3.17) can be integrated to an action S[ϕ] on Ti d+1 where ε is the volume form (2.22) on Ti d+1 .Due to the absence of gradient terms of the form ∂ i ϕ∂ i ϕ this theory is an example of an ultralocal field theory that have been studied in the 60s and 70s (see, e.g., [70] and references therein) and have recently resurfaced in carrollian physics.When we restrict to a scalar this theory is the Ti d+1 analogue of the massive electric Carroll scalar theory [71,72].We will have more to say about this in Section 5.2, where we explicitly consider aspects of the electric Carroll scalar theory on Ti d+1 .Reinstating tensor indices, the solutions of (3.17) are given by where since ϕ is real, we have that ϕ + = (ϕ − ) * .In Section 4.1 we will connect these carrollian fields to the expansion of massive fields in Minkowski space near timelike infinity.But first we will elucidate on the connection to Wigner's massive unitary irreducible representations.

Relation to massive UIRs of the Poincaré group
It is useful to recall Wigner's construction of the massive unitary irreducible representation (UIR) of the Poincaré group [50], and contrast it with the construction of carrollian fields of the previous subsection.For the massive UIR the isotropy subgroup is spanned by rotations and temporal and spatial translations which are generated by ⟨J ab , H, P a ⟩.The inducing representation is then again given by but since H is now in the isotropy subgroup we also specify corresponding to a massive particle in the rest frame with momentum p ± µ = ±(m, 0).The induced representation is then obtained by boosting ϕ ± to a generic momentum frame, where we have defined the coordinates by U : H d → G and U (x) = e x a Ba .This implies and following similar computations as in Section 3.1, yields [P a , ϕ ± (x)] = ∓im sinh(r) xa ϕ ± (x) . (3.24d) The main difference between the massive UIR and the Ti fields is the action of H and P a , which in contrast to (3.15) acts by a phase and does not change the coordinates of the fields (cf.Footnote 2).Indeed, their action yields the Lorentz boosted momentum p ± µ = ±(m cosh(r), −m sinh(r)x a ) in terms of rapidity r and boost direction xa and provide us with the familiar equation Consequently, the action of the Casimir C 2 = −H 2 + P a P a is given by which ensures that the field is on the mass-shell.For fixed m the Casimir is a multiple of the identity, as is necessary for an irreducible unitary representation.This can again be compared with our fields on Ti where we needed to impose the additional equation (3.17) at the end of the construction.
We are now in the position to compare the representation obtained in the last section to the classic Wigner construction recalled above.As is well-known, the representations ϕ ± (x) defined above are both unitary and irreducible.On the other hand, the Ti representation obtained in the last section is also unitary but reducible.Indeed, the latter can be inferred from Schur's lemma for unitary representations and the fact that the Casimir (3.16) is not a multiple of the identity before (3.17) is imposed.More precisely, we will now argue that the Ti representation can be understood as a direct integral of irreducible representations; for a rigorous discussion of this concept see, e.g., Chapter 5 in [73].From equations (3.15) and (3.24) it is clear that the difference between the two representations lies only in the representation of H and P a .Let us therefore start with either one of the irreducible representations ϕ ± (x) from which we construct where we explicitly added the dependence of the Wigner representation on the mass m.A short computation shows that applying the transformations of (3.15) induces the Wigner transformations discussed above.We can therefore decompose a Ti field as a Fourier transform over irreducible representations.Explicitly we have where P Ti µ denotes the action (3.15) on Ti fields and P Wigner µ the action on the unitary irreducible representation (3.24).The reducible representation on the Ti fields is unitary with respect to the measure (2.22) on Ti, where ϕ • ψ is the pointwise inner product associated with the finite-dimensional SO(d) representation.Denoting the measure with respect to which the UIR (3.24) is unitary by where m is the mass of the corresponding representation, the above relations allow to derive This demonstrates again clearly that the novel Ti representation discussed in the previous subsection is an integral over the well-known massive particle states.

Relations to fields in Minkowski space and massive celestial CFT
In this section, we establish connections between the carrollian fields on Ti d+1 and massive fields living in Minkowski space M d+1 , as well as their formulation in terms of massive celestial conformal field theory.

Correspondence with massive fields in Minkowski space
In the previous section we constructed massive carrollian field representations of ISO(d, 1) living on Ti d+1 .We will now show that these precisely arise when considering massive fields in M d+1 in the asymptotic limit to future/past timelike infinity.In particular it will be shown that massive carrollian fields encode massive particle UIRs in a natural way, and we will give their explicit decomposition in terms of creation and annihilation operators.We thus start with Minkowski spacetime M d+1 in cartesian coordinates X µ , in terms of which the Minkowski metric η is given by where η µν = diag(−1, 1, . . ., 1) denotes the components of the Minkowski metric in cartesian coordinates; throughout this section, the indices µ, ν will be used to indicate components in cartesian coordinates.To describe the approach to the timelike infinities i ± it is convenient to introduce a hyperbolic foliation of the inner lightcone emanating from the origin 2) The spatial coordinates x = x i for i = 1, . . ., d are defined in (2.18) and provide an intrinsic chart of the quadric H d characterised by the constraint y 2 = −1, while the coordinate τ is identified with the proper distance from the origin X µ = 0 to any point of the corresponding hyperbolic leaf of this foliation.Note that there is no difference between the indices i, j or a, b used in previous sections, namely they run over the same numbers 1, ..., d.However in this section will reserve i, j for coordinate components and a, b for orthonormal frame components to be defined momentarily.The coordinates y i and x i are formally identical to those we used in (2.20) and (2.21) to write down the carrollian structure on Ti d+1 .Future and past timelike infinity i ± are approached in the limit τ → ±∞, respectively.In the coordinate system (τ, x) the Minkowski metric (4.1) takes the form where h ij is the metric on H d that features in (2.21) and (2.20).In cartesian coordinates the vector fields ζ X with X ∈ iso(d, 1) that generate Poincaré transformations are given by where X µ := η µρ X ρ .In order to express them in the hyperbolic coordinates (τ, x), we introduce the projector h µν onto a single hyperboloid leaf of the foliation at fixed τ .In the ambient Minkowski space in cartesian coordinates, this hyperboloid is defined by where τ is constant, implying that the normal vector is ñµ = η µν X ν .Normalising the normal vector, i.e., requiring that η µν n µ n ν = −1, tells us that where we used (4.2).Hence, the first fundamental form of a hyperboloid leaf embedded in On the other hand, the induced metric on the hyperboloid H d τ at fixed τ = const.in (4.3) is η which we can push forward to the first fundamental form (4.7) as follows where the components of P i µ are explicitly given by Treating all Poincaré transformations (4.4) at once by using the standard parameterisation of an isometry of Minkowski space with A µ and Ω µν = −Ω νµ constants corresponding to translations and Lorentz rotations, we can then evaluate their components in (τ, x) coordinates in the limit τ → ∞.This yields where the last equality in each line is obtained by first writing out the expression in cartesian coordinates followed by the change of coordinates of (2.18); in particular, this means that the parameters A 0 etc. that appear above are still the same constants that appear in (4.10).We recognise the expressions in (4.11) as the components of the Killing vector fields on Ti d+1 given in (2.19).Note also that the Casimir operator can be computed from P µ A µ = ξ τ ∂ τ with the choice that all the translations parameters A µ are equal to one.This yields the expected result This provides the first hint that the carrollian structure of Ti d+1 arises from M d+1 in the limit τ → ±∞.
We now turn to the study of massive fields in M d+1 and their behaviour in the limit τ → ±∞ where, as we shall demonstrate, carrollian fields on Ti d+1 emerge.In particular let us consider symmetric tensor fields ϕ α 1 ...αs (X) carrying totally symmetric and traceless massive spin-s particles.These satisfy the on-shell conditions [74,75] □ − m 2 ϕ α 1 ...αs = 0 , (4.13a) Although these conditions are those appropriate for the free fields that carry one-particle states, the analysis in fact also applies to interacting field theories which satisfy the cluster decomposition principle.Indeed, in that case the interactions are built perturbatively from the free theory, and appear in the Lagrangian through cubic and higher polynomials of the fields [76].The resulting additional terms in the equation of motion do not alter the leading asymptotic behaviour of the fields in their limit towards i ± , which is that of the free fields. 4 In the coordinate system x α = (τ, x i ) given in (4.3) the trace and divergence conditions (4.13b) read η ij ϕ ij... = ϕ τ τ ... , η ij ∇ i ϕ j... = ∇ τ ϕ τ ... , (4.14) while the nonzero Christoffel symbols are given by where the Γ k ij [h] are the components of the Levi-Civita connection on H d in x i -coordinates.Using these, and introducing the shorthand notation a tedious but straightforward computation allows us to write the components of the wave operator acting on ϕ i(k)τ (s−k) in the form 4 Interactions with massless particles that mediate long-range forces change the asymptotic behaviour of massive fields in the limit τ → ∞; see, e.g., the analysis in [77,78] for massive scalar QED.Such interactions are therefore excluded from our analysis.
, and where the covariant derivative and laplacian on H d are denoted ∇ H i and △ H = h ij ∇ H i ∇ H j , respectively.Note that the k spatial indices are also implicitly symmetrised over.To simplify this equation we rescale the field components as such that the equation of motion (4.13a) reduces to with c 1 = c 1 (d, s, k) some constant which will not play any role in the subsequent analysis.From (4.19) we can infer that the asymptotic solution takes the form which is the solution to (∂ 2 τ + m 2 ) φi(k)τ(s−k) = 0 and which we can already recognise as the Casimir equation (3.17).The spatial components ϕ ± i(s) (x) are completely unconstrained while the divergence-free condition in (4.14) completely determines the remaining time components ..is (x) carry the physical degrees of freedom of the corresponding spin-s representation of the little group SO(d).The subleading components of the field can be determined recursively from this free asymptotic data.This together with (3.19) suggests that the spatial components of (4.20) provide a concrete realisation of the massive carrollian fields constructed in Section 3.This will be confirmed by studying their Poincaré transformations.
Let us therefore look at the transformation of the physical asymptotic degrees of freedom φi 1 ...is (τ, x) and demonstrate that their transformation under Poincaré symmetries is exactly that of the massive carrollian fields on Ti d+1 .First note that the transformation laws obtained in Section 3.1 are appropriate for for the tangent space components of the fields, which carry indices a 1 , . . ., a s as in (4.21), while the discussion of fields in Minkowski space so far applied to the coordinate components in hyperbolic coordinates.Locally flat coordinate systems can be reached with the use of vielbeins E A α such that where η αβ represents the metric components in arbitrary curvilinear coordinates x α , while η AB = diag(−1, 1, . . ., 1) is the tangent space Minkowski metric.The tangent space indices A, B = 0, . . ., d split according to A = (0, a), where a the spatial tangent space index.For x α = (τ, x i ) related to cartesian coordinates as in (4.2), we can write the vielbeins that make up the Minkowski metric (4.3) as The Poincaré transformation of the bulk components ϕ A 1 ...As can be written in terms of the Killing vector field (4.10), and with the local Lorentz transformation Λ AB chosen such as to leave the background tetrad expressed solely in terms of the vector fields (4.11) on Ti d+1 , and in agreement with the transformations of the fields constructed in Section 3.1 and given in (3.15).For completeness we also note that the transformation of the coordinate components can be fully covariantised on Ti d+1 by adding trivial time-components to the tensor φ, namely where n = ∂ τ is the carrollian vector field of (2.20).
We can summarise the correspondence we just established between massive fields ϕ α 1 ...αs on M d+1 and carrollian fields φa 1 ...as on Ti d+1 through the following extrapolate dictionary where the limit is asymptotic in the sense that the carrollian field φ still depends on τ through the phase factors e ±imτ .

Plane wave basis and reconstruction algorithm
It is useful and informative to relate the above considerations to the standard plane wave decomposition of the massive field in cartesian coordinates where The sum is over all independent polarisations σ and where the polarisation tensor ϵ σ µ 1 ...µs is totally symmetric and traceless, and further satisfies the transversality condition k µ ϵ σ µ...µs (k) = 0, cf.(4.13).The modes a σ k obey the canonical commutation relations We will evaluate (4.33) in the limit τ → ∞ in order to extract an expression for the carrollian field (4.21).In order to achieve this we once more turn to the hyperbolic coordinate system defined via X µ = τ y µ with (cf.(2.18)) and we similarly parameterise the massive momentum as such that (4.33) can be rewritten as The asymptotic limit τ → ∞ can be evaluated by stationary phase approximation as done for instance in [78].In the parametrisation (4.35)-(4.36)we have such that the critical point of (4.37) at large τ is located at The determinant of the second derivative of (4.38) at the critical point is given by Taken together, the stationary phase approximation of (4.37) yields We are actually interested in the spatial components in hyperbolic coordinates (τ, y), obtained from the cartesian components given above through the jacobian transformation5 The degrees of freedom of the massive carrollian field sit in the purely spatial components, thus given by where we have defined the spatial polarisations Note that there are exactly as many independent polarisations σ as there are independent totally symmetric and traceless tensors ϵ i 1 ...is , which is just another way to say that ϵ σ i 1 ...is carries the spin-s irrep of SO(d).The leading τ s− d 2 behavior of (4.43) is indeed the one anticipated in (4.18) for k = s, and comparison with (4.20) allows us to identify the carrollian degrees of freedom This result makes fully explicit the previous statement that the massive carrollian fields essentially are the particle degrees of freedom, up to the change of basis given in (4.26).On the one hand the asymptotic leaf H d of the hyperbolic foliation (4.3) is explicitly identified with the massive momentum shell through the critical point y = k, while the spin-s representation of the little group SO(d) is manifestly carried by the spatial polarisation tensor ϵ σ i 1 ...is .We will close this section by noting that the relations (4.45), when inverted such as to express a σ k in terms of the components of ϕ + i 1 ...is and upon replacement back into the original plane wave expansion (4.33), provide a reconstruction algorithm for the relativistic bulk fields ϕ µ 1 ...µs (X) from its asymptotic value ϕ i 1 ...is (y) near timelike infinity.Explicitly for the scalar field this straightforwardly yields This is the timelike analog of the Kirchhoff-d'Adhémar formula [79,80].From a carrollian perspective this is the Ti analog of what has been considered for null infinity in Section 2.2.2 of [15].

Relation to massive conformal primary wavefunctions
Having established the relation between plane wave creation and annihilation operators and the fields ϕ ± (x) transforming under a representation of Ti at fixed Casimir, we can straightforwardly relate the latter to celestial CFT variables.In the following we rely on the discussion in [81] and restrict ourselves to the case of a scalar field for simplicity.
In the celestial CFT context, one considers the following massive conformal primary basis of solutions to the free wave equation Here, ⃗ w ∈ R d−1 is a point on the boundary of H d and G ∆ ( k; ⃗ w) is the bulk-to-boundary propagator on H d that can be written as where Note that in (4.47) one trades the d spatial momentum components specifying a plane wave solution of given mass for a weight ∆ and d − 1 points on the boundary of the hyperboloid, thus staying with the same number of data.It can be shown that one needs to set ∆ = d/2 + iν with ν ∈ R in order to have delta-function normalisable modes [81].
Instead of expanding a solution to the massive Klein-Gordon equation in terms of plane waves and corresponding creation and annihilation operators a k , a † k as in (4.33), one expands the field into conformal primary waves ϕ ± ∆ (X µ ; ⃗ w) and celestial operators with inverse relation where µ(ν) is a normalisation factor [66,81].
Since the fields on Ti d+1 are proportional to the plane wave creation and annihilation operators, the relation is now obvious.Explicitly, we find from (4.45) and We have summarised these interrelations between the momentum, carrollian and celestial points of view in Figure 4, which can be contrasted with the analogue for null infinity in [15].

Carrollian energy-momentum tensor and conservation laws
In this section, we provide an intrinsic definition of a carrollian energy-momentum tensor (EMT) on Ti d+1 .We then explicitly construct the EMT of a massive scalar in two complementary ways: first by pushing the lorentzian EMT from Minkowski to timelike infinity using the techniques of Section 4.1, and then by varying the lagrangian of an electric Carroll scalar field theory.Finally, we discuss the relation between additional conserved charges corresponding to the Ti-supertranslations (2.26) and BMS charges as well as their connection to soft graviton theorems.

Intrinsic description
We begin with a discussion carrollian EMTs from a variational perspective (see, e.g., [38,56,[82][83][84], which allows us to derive the general properties of carrollian EMTs.At the same time, this procedure provides an explicit recipe for computing the EMT of a given theory by coupling it to an arbitrary carrollian background.We recall that in a relativistic theory, the EMT may be extracted by coupling the theory to a curved lorentzian background and varying with respect to the metric, and further arises as the conserved current associated with diffeomorphism symmetry.In other words, given a relativistic field theory coupled to a lorentzian background with metric g µν described by an action S R [φ I ; g µν ], where φ I abstractly captures the field content, the variation takes the form δS R [φ I ; where T µν R is the relativistic EMT, and E I R are the equations of motion for the fields φ I .The symmetry of g µν immediately tells us that T µν R is symmetric, while diffeomorphism invariance of the action leads to conservation of T µν R when E I R = 0.If we denote the Levi-Civita connection of g µν by ∇, the metric transforms as under an infinitesimal diffeomorphism generated by ζ µ , and on-shell invariance of S R implies that where we used symmetry of T µν R and threw away boundary terms arising from integration by parts.Since the above must vanish for arbitrary ζ µ , we must have that ∇µ T µν R = 0 . ( This is the diffeomorphism Ward identity for a relativistic field theory.For a carrollian theory with generic field content Φ I for I some abstract label, the same principle applies: coupling it to an arbitrary carrollian geometry allows us to extract the carrollian EMT.More precisely, if the carrollian background is given in terms of the carrollian structure (n µ , q µν ) with inverses (τ µ , γ µν ) the variation of the action S[Φ I ; τ µ , q µν ] can be written as [38,56,82,84] where the measure6 is e = det(τ µ τ ν + q µν ), and where T µ is the energy current, T µν = T νµ is the momentum-stress tensor, and E I represents the equations of motion for the fields Φ I .
Computing the second variation under local carroll boosts, δ C (δ C S) = 0, and assuming that Φ I is Carroll boost invariant, the fact that τ µ transforms as in (2.3) implies that T µν is not boost invariant: δ C T µν = −2T (µ γ ν)ρ λ ρ , while the energy current T µ remains invariant (see [56] for more details).Furthermore, requiring that S[Φ I ; τ µ , q µν ] is invariant under local Carroll boosts (2.3) yields the boost Ward identity This relation is true off-shell 7 , which means that we may without loss of generality express the energy current as where we used (2.2).The quantity E is a boost invariant scalar which we can think of as an energy.Finally, when E I = 0, i.e., when the Φ I are on-shell, a calculation analogous to (5.2) shows that diffeomorphism invariance gives rise to the 'conservation' equation where we defined the EMT T µ ν = τ ν T µ + T µρ q ρν , ( which is boost invariant by virtue of the boost Ward identity (5.5).This means that, given a carrollian Killing vector ξ µ satisfying (2.6), the contraction of (5.7) with ξ µ gives where we used (2.6).Hence, the combination j µ = T µ ν ξ ν is a conserved current.In what follows, we shall assume the existence of an adapted and torsion-free carrollian connection ∇ as discussed around Eq. (2.5), in terms of which the diffeomorphism Ward identity (5.7) becomes while the current conservation (5.9) takes the form which is identical in form to the corresponding relation in lorentzian theories.
To define the EMT on Ti d+1 and derive the currents for a given theory, we therefore should couple the theory to a generic carrollian geometry and vary the background as detailed above, and then specialise to the case of Ti d+1 once the general expressions have been found.Alternatively, we may construct the conservation laws directly on Ti d+1 using its carrollian structure (2.20).
Turning (5.11) on its head, we may define the EMT T µ ν on Ti d+1 by demanding that its contraction with a Killing vector ξ µ on Ti d+1 j µ = T µ ν ξ ν , (5.12) leads to a conserved current in the absence of sources As discussed in Section 2.4, the isometries of Ti d+1 include supertranslations with ξ S = S(x)∂ τ which in particular contain the Poincaré translations.For these supertranslations, we obtain which can be satisfied only if We recognise the second of these as the carrollian boost Ward identity (5.5).It is straightforward to check that restricting to the Poincaré translations would lead to the same conclusion.The Killing vectors associated with boosts and rotations are of the form ξ = ξ a (x)∂ a , which yields where we defined ∇ H a as the covariant derivative on H d with connection components Γ a bc .If ξ b is a Killing vector on the hyperboloid, i.e., a boost or a rotation, then only the antisymmetric part of T a b will remain.The above equation can then be written where we used the metric on the hyperboloid H d to raise and lower the spatial indices.
Taking ξ a to be the generator of a rotation ξ a J bc = x b δ a c − x c δ a b allows us to write which leads to when plugged back into (5.17).In order for this to vanish for all Killing vectors, we have to demand Taken together, we find that the carrollian EMT is of the form with T ab symmetrical, and furthermore obeys which we recognise as the conservation equation (5.10).Note in particular that these conditions are independent of the choice of the undetermined part of the connection.Using the conserved current (5.12) and the invariant top-form (2.22), we can define conserved charges as integrals over the hyperboloid (5.23) In particular, we find for supertranslations and for boosts and rotations where the ξ a are the vectors ξ Ba and ξ J ab given in (2.19).

An example: the energy-momentum tensor of massive scalars
As an illustration of the above, we will consider the carrollian EMT associated with massive real scalars.We obtain it in two ways, first by pushing the corresponding relativistic EMT from Minkowski space M d+1 to Ti d+1 , and then from a carrollian action principle.We thus start by considering a free massive scalar field in Minkowski space.The corresponding EMT is given by (5.26) We want to push this tensor to timelike infinity to obtain an example of a carrollian EMT.
In hyperbolic coordinates we have ) ) where we used the rescaled field φ = τ d 2 ϕ defined in (4.18).At leading nontrivial order we have where we used the solutions of the leading equations of motion (4.21), and where ϕ − = (ϕ + ) * since we are dealing with real scalars.This carrollian EMT can be shown to satisfy the carrollian conservation equations (5.22).
Although we started with the EMT of a free theory in (5.26), the resulting carrollian EMT (5.28) will not change if a potential is included.As discussed below (4.13), the leading order field in a large τ expansion is still given by the free field.Furthermore, contributions coming from the potential will fall off faster than τ −d so that the leading order terms in (5.28) stay unaltered.
Using the explicit form of the EMT one can establish that the charges (5.23) with the EMT (5.28) form a representation of the symmetry algebra of Ti.The commutator of the carrollian fields (5.33) follows from the canonical commutation relations (4.34) and the identification (4.45); the delta function δ(d) (y − y ′ ) is associated to the volume form on the hyperboloid in y a coordinates (2.22).It follows then that which is indeed a representation of the symmetry algebra on Ti. 8More generally, the canonical commutation relations imply the following algebra of the carrollian EMT This algebra is arguably quite different from the EMT algebra of a lorentzian QFT as determined originally in [85][86][87].In particular, the vanishing of the [ T τ τ (y), T τ τ (y ′ )] commutator can be regarded as hallmark of carrollian symmetry [71].In a lorentzian field theory, the EMT algebra is universal, except for the commutator of purely spatial components, and can be derived by coupling the theory to a curved background and subsequent variation of the metric.It would be interesting to similarly analyse the universality of (5.35) and investigate the possibility of central terms by applying an analogous procedure using the background carrollian structure.

Relation to BMS charges and soft graviton theorems
In previous subsections we have shown that conserved charges are associated to Killing symmetries of Ti d+1 .In four spacetime dimensions (d = 3) it turns out that BMS symmetries which consist of supertranslations [7][8][9] and Lorentz transformations can be identified as a subset of these symmetries of Ti 4 .In fact the modern view is that a single BMS symmetry group acts simultaneously on all asymptotic regions of asymptotically flat spacetimes [54,[88][89][90][91][92][93], with a single function f (Ω) valued over the celestial sphere CS 2 to parametrise supertranslations.In particular supertranslations form a subset of the Ti-supertranslations S(x) which are determined from f (Ω) through the formula [54,88] where G(x, Ω) is a 'boundary-to-bulk' propagator satisfying the gauge-fixing equation and chosen in such a way that f (Ω) is the asymptotic boundary value of S(x) along the asymptotic boundary ∂H 3 ∼ = CS 2 .Hence with this particular choice of S(x) the corresponding charges (5.24) generate the BMS supertranslations of the massive carrollian fields (i.e., massive particles), while the charges (5.25) generate their Lorentz transformations.
It is by now a well-known result that Weinberg's soft graviton theorem [51] is the Ward identity associated with supertranslation symmetry [52][53][54].We can sketch the proof of this result in the following way. 9Working with a theory of gravity and interacting matter whose phase space is such that there exist truly conserved BMS charges Q(i 0 ) at spatial infinity, the latter can be re-expressed as a sum of a BMS fluxes going through I ± and a remaining BMS charge at i ± , (5.38) Invariance of the S-matrix under BMS symmetries, The soft graviton theorem is then obtained by acting with F (I ± ) and Q(i ± ) on in and out states.More specifically each flux can be decomposed into so-called hard and soft contributions F (I ± ) = F hard (I ± ) + F soft (I ± ).The action of the soft flux F soft (I + ) is to explicitly add a soft graviton to the outgoing states, while the action of F hard (I + ) and Q(i + ) generate the BMS transformations of the massless and massive external particle states, respectively.The same holds for the operators acting on the in states.Writing this explicitly, manifestly reproduces Weinberg's soft graviton theorem [52][53][54].
Having laid down these facts it should not come as a surprise that the BMS supertranslation charges Q(i ± ) entering the (re-)derivation of Weinberg soft graviton theorem in the presence of massive scalar particles [54] are nothing but the Ti-supertanslation charges (5.24) with the particular restriction (5.36) on the symmetry parameter S(x).Indeed upon using the carrollian EMT for massive carrollian scalar fields given in (5.28) together with the relation (4.45) between carrollian fields and particle operators, the charge (5.24) takes the explicit form which exactly agrees with Eq. ( 36) in [54] up to a choice of normalisation.This demonstrates that Ti 4 provides the natural asymptotic structure for massive fields even when it comes to fully interacting theories.This discussion should hold for massive spinning particles as well although this has not been explicitly studied in the literature.

Discussion and outlook
Motivated by flat space holography we have provided a carrollian description of massive particles.We have done this from an intrinsic perspective, as in Section 2, and as a late time limit of Minkowski space, as in Section 4. This provides an extrapolate dictionary for (symmetric) massive integer spin s fields in generic dimension.We have also related this carrollian description to the plane wave, momentum and celestial perspectives, cf. Figure 4.This work also shows that there are interesting applications for Carroll physics beyond the more extensively studied flat case.In addition, we have provided an intrinsic definition of the carrollian energy-momentum tensor on Ti d+1 and discussed the associated Ti-supertranslation charges along with their relation to soft gravitation theorems.This work opens various interesting avenues for further exploration: Massive scattering amplitudes A natural extension of the present work is the study of correlation functions of massive carrollian fields on Ti, and their relation to the standard scattering amplitudes of massive particles.As a simple example we can consider the free 1-1 scattering of a massive scalar particle, with S-matrix element simply given by the Lorentz invariant inner product where the last equality follows from the natural parameterisation of the massive momenta given in (3.25).This simple amplitude can be viewed as a two-point function for the corresponding carrollian field with N some normalisation factor, since for any of the Poincaré transformations δϕ given in (3.15) it can be shown to satisfy the integrated Ward identity .3)This correspondence between massive scattering amplitudes and carrollian correlation functions on Ti should hold generally as it is a simple consequence of the covariance of the S-matrix together with the direct relation between massive particle states and carrollian fields discussed in this work.Note that scattering amplitudes are defined on the momentum mass-shell H d , and are therefore the local observables of a codimensionone theory "dual" to field theory on M d+1 .Furthermore the corresponding carrollian correlators are simple uplifts of the amplitudes from H d to Ti d+1 .Nevertheless it will be of high interest to study whether this new perspective yields new ways to constrain and construct massive scattering amplitudes.We point out that the analogous connection between massless scattering amplitudes and correlation functions of carrollian conformal fields at null infinity I has been the subject of numerous recent works [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24].
General scattering More generally, this work also opens the possibility to study general scattering between massive and massless particles from a carrollian perspective.The massive and massless carrollian fields seem to live on different asymptotic boundaries, but a careful analysis could make it possibly to match them, see, e.g., [93] and references therein for work in this direction.Another possibility would be to use the embedding space picture we discuss in Section 2.3 where spatial, timelike and null infinity embed in a larger connected space.

Superrotations and subleading soft graviton theorems
The question naturally arises whether Ti 4 also captures the extended version of the BMS group that includes Virasoro superrotations in addition to supertranslations [96,97].Similarly the subleading soft graviton theorem [98] and associated loop corrections [99,100] are also known to derive from the Ward identities of superrotation symmetries [54,94,95,[101][102][103]. However the group of carrollian isometries of Ti 4 does not contain symmetries that could be identified with Virasoro superrotations (beyond the standard Lorentz transformations, cf., also our discussion at the end of Section 2.4).To better understand the origin of Virasoro superrotations it is useful to project from Ti 4 down to its hyperbolic base H 3 also known as euclidean AdS 3 .In that case it is known that Virasoro symmetries arise as asymptotic symmetries of three-dimensional gravity with AdS 3 asymptotics [104].Although the latter are conformal isometries of the celestial sphere ∂H 3 ∼ = CS 2 , away from this asymptotic boundary they do not leave the metric on H 3 invariant.This is no issue however since the metric in the bulk of H 3 is a dynamical gravitational field allowed to fluctuate.This suggests that Virasoro superrotations arise at Ti 4 when the metric (2.20) is allowed to fluctuate away from ∂H 3 ∼ = CS 2 , which would appear natural in a theory of gravity with flat asymptotics.
In fact the only kind of fluctuations needed to witness the appearance of Virasoro superrotations are locally pure diffeomorphisms.Indeed an implementation of the above ideas has been presented in [105].In short the description of superrotation symmetries and charges requires to go beyond isometries of Ti 4 by allowing for metric fluctuations, and therefore goes beyond the kinematic framework presented here.
To go beyond the kinematical framework of this work one can "gauge" the Ti 4 symmetries, i.e., builds theories which allow for curved and dynamical geometries.This is the analogue of gauging Minkowski space (often referred to as "gauging the Poincaré algebra") which leads to Einstein gravity in the first order formalism.For Ti 4 (or equivalently AdS Carroll) this was done in [58] (see also [57]) and leads to a carrollian theory of gravity. 10An asymptotic analysis of this theory [108] leads to the expected BMS 4 symmetries and also shows that it allows for superrotations.It would be interesting to see whether this gauged Ti 4 theory naturally arises at timelike infinity if gravity in the bulk asymptotically flat spacetime is dynamical.

Detector operators
The integral along null infinity of (weighted) components of the EMT in asymptotically flat spacetime can be interpreted as detector operators measuring properties of the outgoing flux of massless particles in a given state. 11Here we will argue that the carrollian EMT discussed in Section 5 can be used to construct idealised detectors for massive particles.The application of this operator on the final state (6.4) yields which as expected gives the sum of the energies of the particles whose spatial momenta were aligned to n.The operator E(n) provides a massive analogue to the well-known energy-flow operator counting the total energy carried by massless particles along the direction n; see, e.g., [113] for applications of this operator to e + e − annihilation.Note that it is also the charge (5.24) associated with a Ti-supertranslation S n (y) = − 1 + y 2 δ (d−1) (n y − n).The hallmark equation of carrollian physics, i.e., the vanishing of the [ T τ τ (y), T τ τ (y ′ )] commutator, implies that the energy-flow operator commutes [E(n), E(n ′ )] = 0 , (6.10) i.e., the energy flow in different directions can be measured independently from each other.Finally, T τ τ (y) itself measures the energy as a function of the spatial momentum.
The above only presents one example of conceivable detector operators for massive particles, the design of which depends on the questions to be answered in a given thought experiment.Although arguably not as rich due to the absence of collinear and soft divergences, it should be clear that the carrollian EMT and its algebra can be viewed as providing the basic building blocks for detector operators for massive particles.
Curved space Our discussion was based on a fixed Minkowski background and it would be interesting to generalise to curved backgrounds.In the following, let us present a simple argument that the results of Section 4.1 should carry over.
Assuming that the Ashtekar-Hansen definition of asymptotic flatness at spatial infinity [47] in four dimensions can be adapted to Ti, we require the existence of an unphysical, conformally related metric gαβ = Ω 2 g αβ that is continuous at time-like infinity.In contrast to null infinity, the conformal factor is required to obey the properties Ω| i + = 0, ∇α Ω| i + = 0, ( ∇α ∇β Ω + 2g αβ )| i + = 0 .(6.11) Similar to the relation between the Ashtekar-Hansen construction and hyperbolic coordinates near spatial infinity (see Appendix B in [114] for a comprehensive discussion) we relate the conformal factor to a time coordinate as Ω = τ −2 .We can use this to perform an asymptotic analysis of the wave equation for a scalar field in four dimension.Writing the wave equation in terms of the unphysical metric yields 0 = (g αβ ∇ α ∇ β − m 2 )ϕ = (Ω 2 gαβ ∇α ∇β − 2Ωg αβ ∇α Ω ∇β − m 2 )ϕ.(6.12) Replacing the inverse metric using the third equation of (6.11) one finds that the Laplacian is projected onto −∂ 2 τ to leading order in Ω.Consequently, one finds the asymptotic equation (−∂ 2 τ − m 2 )ϕ + O(Ω 1/2 ) = 0, (6.13) so that the asymptotic solutions are still given by the Ti fields (4.21) in agreement with the strictly flat case.
Clearly, this analysis depends on a bona fide extension of the Ashtekar-Hansen analysis to time-like infinity by analytic continuation.A more thorough discussion of the general curved case would therefore require a similar analysis of appropriate boundary conditions at time-like infinity along the of e.g., [93].
We end by noting that the work [115] demonstrated that generic asymptotically flat solutions to Einstein gravity can be compactified so that they give rise to curved versions (in the Cartan sense) of Ti at asymptotically late times.It would be interesting to make closer contact with this analysis and similarly interpret the late-time behaviour of fields on a curved bulk spacetime in terms of fields on these curved analogues of Ti.

Figure 2 .
Figure 2. The blowup of future timelike infinity i + in (d + 1)-dimensional flat spacetime M d+1 is the carrollian space Ti d+1 ∼ = AdSC d+1 , which has the structure of a trivial line bundle over d-dimensional hyperbolic space H d .The Penrose diagram above includes the hyperbolic foliation of flat spacetime, where red lines correspond to copies of H d , while blue lines correspond to d-dimensional de Sitter spaces. )

Figure 4 .
Figure 4. Three different perspectives on massive states in flat spacetime and their interrelations: carrollian (top), momentum space (left) and celestial (right).

7 )
As a concrete example, consider the final state of a scattering experiment consisting of n particles of identical mass m|n⟩ = |p 1 , . . ., p n ⟩ = a † p 1 . . .a † pn |0⟩ = m 1−d/2 2(2π) d/2 n ϕ − (y 1 ) . . .ϕ − (y n ) |0⟩ .(6.4)From the normal-ordered carrollian energy density operator: T τ τ (y) := −2m 2 ϕ − (y)ϕ + (y) ,(6.5)one can construct an operator that just measures the total mass of the final state− H ε (d) : T τ τ (y) : |n⟩ = n m |n⟩ , (6.6) or the total energy − H ε (d) 1 + y 2 : T τ τ (y) : |n⟩ = These are of course nothing but the supertranslation charge for S = −1 and global time translation S = − cosh r = − 1 + y 2 .At particle colliders what is typically measured is the total energy density which leaves along a given spatial direction n from the origin of the detector.This implies that the spatial momentum of a detected particle was proportional to n, namely p = ∥p∥n.Now recall that the asymptotic leaf H d is identified with the momentum mass shell of the particles through the critical point equation(4.39).Using the natural parametrisation p µ (y) = m( 1 + y 2 , y) that follows from (4.35)-(4.36),we equivalently have y = ∥y∥n for any of the detected particles, and the corresponding total energy density left along the direction n is therefore obtained by integrating : T τ τ (y) : over the radial coordinate ∥y∥ on the hyperboloid, .26)where S(x) is an arbitrary smooth function on H d .In particular, the vector fields ξ Pa and ξ H in (2.19c)-(2.19d)associated with Poincaré translations are special cases of (2.26).