Galilean first-order formulation for the non-relativistic expansion of general relativity
GGalilean first-order formulation for the non-relativistic expansion of general relativity
Dennis Hansen, Jelle Hartong, Niels A. Obers,
3, 4 and Gerben Oling Institut f¨ur Theoretische Physik, ETH Z¨urich, Wolfgang-Pauli-Strasse 27, 8093 Z¨urich, Switzerland School of Mathematics and Maxwell Institute for Mathematical Sciences,University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden
We reformulate the Palatini action for general relativity (GR) in terms of moving frames thatexhibit local Galilean covariance in a large speed of light expansion. For this, we express the actionin terms of variables that are adapted to a Galilean subgroup of the GL ( n, R ) structure group of ageneral frame bundle. This leads to a novel Palatini-type formulation of GR that provides a naturalstarting point for a first-order non-relativistic expansion. In doing so, we show how a comparisonof Lorentzian and Newton–Cartan metric-compatibility explains the appearance of torsion in thenon-relativistic expansion. INTRODUCTION
In recent years the study of the non-relativistic regimeof general relativity (GR) has gained renewed interest.This has been driven to a large extent by new insights[1–3] in
Newton–Cartan geometry [4, 5], which in thenon-relativistic limit takes the place of the Lorentziangeometry of GR. Building on earlier work [6, 7], a ge-ometric description of the non-relativistic expansion ofgeneral relativity in inverse powers of the speed of light c was recently obtained in Refs. [8–10]. This paves theway for a covariant formulation of the post-Newtonianexpansion. More generally, covariant formulations in-volving Newton–Cartan-type geometries prominently ap-pear in recent developments in non-relativistic field the-ory [11–14], non-relativistic string theory and limits ofthe AdS/CFT correspondence [15–17], along with manyother areas that exhibit non-relativistic physics.In this work, we focus on the 1 /c expansion of GR,which has been shown to lead to a modification ofthe original notion of Newton–Cartan (NC) geometry,known as type II torsional Newton–Cartan (TNC) ge-ometry [9, 10], as the correct framework for a covari-ant action of non-relativistic gravity. When coupled toa point particle, the equations of motion of this actionlead to the Poisson equation of Newtonian gravity. Inaddition, for appropriate matter sources, the theory gen-eralizes Newtonian gravity since it includes the effectsof gravitational time dilation due to strong gravitationalfields [8, 10, 18]. When time is absolute, type II TNCgeometry reproduces the usual NC geometry.To set the stage, we briefly review the non-relativisticexpansion of GR in powers of 1 /c following Ref. [10]. Wecan make the factors of c in the metric explicit by intro-ducing the ‘pre-non-relativistic’ (PNR) parametrization g µν = − c T µ T ν + Π µν , g µν = − c V µ V ν + Π µν . (1)The resulting variables can be expanded in powers of 1 /c and are all of order O ( c ). The leading terms in the expansion of the metric and its inverse are due to T µ , Π µν . (2)This timelike vielbein and inverse spatial ‘metric’ satisfy T µ Π µν = 0, which means that Π µν has corank one.Before we expand these variables in 1 /c , they sim-ply provide a parametrization of the Lorentzian metric.However, at leading order, T µ and Π µν define a NC ge-ometry, which is modified into a type II TNC geome-try by the subleading corrections. As such, the PNRparametrization (1) is a convenient way of recasting theLorentzian metric variables of GR in such a way that theappropriate non-relativistic geometry appears naturallyin the expansion.Next, we rewrite the Einstein–Hilbert action of GR, S EH = c πG N (cid:90) M L (cid:48) EH √− g d n x, L (cid:48) EH = R, (3)in terms of the PNR variables (1). Here, R is the Ricciscalar of the Levi-Civita connection, which is the uniquetorsion-free connection that preserves the Lorentzianmetric g µν under parallel transport, as is convenient fordescribing the degrees of freedom of GR. However, toobtain a connection that is compatible with the non-relativistic geometry arising from our expansion, it isconvenient to use the PNR connection C ρµν = − T ρ ∂ µ T ν + Π ρσ ∂ ( µ Π ν ) σ −
12 Π ρσ ∂ σ Π µν , (4)which leads to a covariant derivative under which both ofthe leading PNR variables (2) are covariantly constant.The Newton–Cartan structure arising from these vari-ables at leading order in the 1 /c expansion is then co-variantly constant with respect to the leading order termof this covariant derivative. Note that the connection (4)has nonzero torsion.Up to a total derivative, the Einstein–Hilbert La-grangian then takes the form L (cid:48) EH = c µν Π ρσ T µρ T νσ + Π µν (C) R µν − c V µ V ν (C) R µν , (5) a r X i v : . [ h e p - t h ] D ec where T µν = ( dT ) µν = ∂ µ T ν − ∂ ν T µ and (C) R µν is the Riccitensor corresponding to the connection (4). Note thatwe have factored out c √− g/ (16 πG N ) of the Lagrangianfor brevity, and √− g is of order c , so this prefactor is oforder c . The parametrization (5) is the starting point forthe large speed of light expansion performed in Refs. [9,10], from which we obtain a truncated action at eachorder in 1 /c . Such an action has been constructed upto second subleading order, leading to the theory of non-relativistic gravity discussed above, but the expansionbecomes increasingly cumbersome at higher orders.In this Letter, our aim is to introduce the pre-non-relativistic form of the first-order Palatini action of GR,which simplifies the computation of the 1 /c expansion athigher orders. In addition, it is expected to simplify thestudy of a wide range of physical applications, for exam-ple by making the construction of boundary charges moreaccessible (see for example [19]), enabling the coupling tofermion fields, and simplifying Kaluza–Klein reductions.Furthermore, we provide a new perspective on the ge-ometric interpretation of the PNR parametrization us-ing moving frames , which clarifies the appearance oftorsion and the local Galilean symmetries associated toNewton–Cartan geometry. By embedding both the localLorentz symmetry of GR and the local Galilean sym-metry of NC geometry inside the GL ( n, R ) frame bun-dle and its associated general (linear) affine connections,we can translate between the corresponding notions oftorsion and metric-compatibility. This affine perspectivealso naturally connects to the ‘triality’ between the usualmetric formulation of GR and its equivalent formulationsin terms of torsion or non-metricity [20, 21]. AFFINE AND LORENTZIAN CONNECTIONS
In the following, we will frequently use moving frames,which are also commonly known as vielbeine, to describethe geometry of the tangent bundle. For a spacetime M of dimension n , we denote by E A = E Aµ dx µ , Θ A = Θ µA ∂ µ , E A (Θ B ) = δ AB , (6)a set of vielbeine and their duals. Additionally, we intro-duce a connection Ω AB = Ω µAB dx µ and its associatedcovariant derivative D , which acts on frame tensors as DX AB = dX AB + Ω AC ∧ X CB − Ω CB ∧ X AC . (7)At this point, we take Ω AB to be a general (linear) affineconnection, which takes values in gl ( n, R ). The torsiontwo-form T A of the connection is given by T A = dE A + Ω AB ∧ E B . (8) Additionally, under local GL ( n, R ) transformations Λ AB ,the connection and a frame tensor X AB transform as δ Ω AB = − d Λ AB + Λ AC Ω CB − Λ CB Ω AC , (9) δX AB = Λ AC X CB − Λ CB X AC . (10)Finally, the frame bundle connection Ω AB can be relatedto a connection Γ ρµν acting on tensor products of the tan-gent and cotangent bundle by requiring the vielbeine tobe parallel with respect to the sum of these connections,0 = D µ E Aν = ∂ µ E Aν − Γ ρµν E Aρ + Ω µAB E Bν . (11)This relation is also known as the ‘vielbein postulate’.So far, none of the above requires the existence ofa metric. Introducing a Minkowski metric η AB on theframe bundle breaks the GL ( n, R ) transformations of ageneral frame bundle down to the SO (1 , n −
1) Lorentztransformations that leaves it invariant. The Lorentz al-gebra also corresponds to the local symmetries of theassociated Lorentzian tangent bundle metric g µν = η AB E Aµ E Bν . (12)The metric η AB may not be covariantly constant for ageneral affine connection, which we can measure using Q AB = − Dη AB / AB + Ω BA ) / ( AB ) . (13)We therefore refer to Q AB as the Lorentzian non-metricity tensor . Note that we have lowered one of theindices on the connection using η AB , which allows us toconsider its (anti)symmetrization, so that we can splitΩ AB = Ω [ AB ] + Q AB , (14) T A = dE A + Ω [ AB ] ∧ E B + Q AB ∧ E B . (15)We can then use the torsion equation (15) to solve forΩ [ AB ] in terms of the non-metricity Q AB and torsion T A ,Ω AB = (cid:98) Ω AB + K AB + L AB . (16)This expresses a general gl ( n, R ) connection Ω AB interms of the Levi-Civita connection, which we denote by (cid:98) Ω AB = dE [ A (cid:0) Θ B ] , Θ C (cid:1) E C − dE C (Θ A , Θ B ) E C , (17)as well as the contorsion K AB and disformation L AB , K AB (Θ C ) = 12 T C (Θ A , Θ B ) − T [ A (Θ B ] , Θ C ) , (18) L AB (Θ C ) = 12 Q AB (Θ C ) − Q C ( A (Θ B ) ) . (19)The decomposition (16) implies the well-known fact thatthe Levi-Civita connection (cid:98) Ω AB is the unique torsion-freeconnection that preserves the Lorentzian metric η AB .Following Equation (14), we have thus seen that in-troducing a frame metric such as η AB naturally splits anaffine connection Ω AB into a part Ω [ AB ] that preservesthe metric and a part Q AB = Ω ( AB ) that does not. Wewill consider the Galilean version of this decompositionof a general gl ( n, R ) connection in the following. PALATINI ACTION AND SHIFT
We now turn to general relativity, where our startingpoint is the frame formulation of the Palatini action, S Pal [ E, Ω] = 12 κc (cid:90) M η AB ∗ ( E A ∧ E C ) ∧ R BC , (20)with κ = 8 πG N c − . This action contains both the viel-beine E A and a gl ( n, R ) connection Ω AB as variables.The latter appears through the curvature two-form R AB = d Ω AB + Ω AC ∧ Ω CB . (21)Furthermore, the Hodge dual ∗ leads to a ( n − ∗ ( E A ∧ E C ) = η AD η CD ( n − (cid:15) D ··· D n E D ∧ · · · ∧ E D n , (22)so that the total integrand of the action (20) is an n -form.Although the Palatini action is formulated using theMinkowski metric η AB , we actually do not need toassume that the connection variable Ω AB is metric-compatible or torsionless. Starting from a fully general gl ( n, R ) connection, the equations of motion lead toΩ AB = (cid:98) Ω AB + η AB Z. (23)The one-form Z is arbitrary but drops out of the actionupon substituting the solution, and we can remove thisambiguity using a Lagrange multiplier for a constraint,see for example [22, 23]. With that, out of all possible gl ( n, R ) connections, the Palatini action then uniquelyselects the Levi-Civita connection (17). Substituting thissolution in the action reproduces the Einstein–Hilbertaction (3).Our goal is now to identify a first-order formulationof the PNR parametrization of the Einstein–Hilbert ac-tion (5) in terms of the adapted connection C ρµν thatwas mentioned in the Introduction. This can easily beachieved using the linear field redefinitionΩ AB = (cid:101) Ω AB + S AB , (24)where the ‘shift’ parameter S AB is a particular functionof the vielbeine and their derivatives. The on-shell valueof the new connection variable (cid:101) Ω AB is then given by (cid:101) Ω AB (cid:12)(cid:12)(cid:12) on-shell = (cid:98) Ω AB − S AB . (25)As a result, even though the Palatini action (20) im-plies that the on-shell value of Ω AB corresponds to theLevi-Civita connection, an alternative connection can beobtained on shell from (cid:101) Ω AB using a suitable shift S AB .In the action (20), the field redefinition (24) leads to S [ E, (cid:101) Ω] = 12 κc (cid:90) M η AB ∗ ( E A ∧ E C ) (26) ∧ (cid:16) ˜ R BC + S BD ∧ S DC + ˜ DS BC (cid:17) . This ‘shifted’ action now depends on (cid:101) Ω AB , and throughthe field redefinition (24) it is equivalent to the Palatiniaction (20). In the following, we will identify a suit-able shift S AB to obtain the frame equivalent of the de-sired PNR connection (4). To show that the resultingaction (26) is equivalent to the PNR parametrization (5)of the Einstein–Hilbert action, it is useful to rewrite it as S [ E, (cid:101) Ω] =12 κc (cid:90) M η AB ∗ ( E A ∧ E C ) ∧ (cid:16) ˜ R BC − S BD ∧ S DC (cid:17) + d (cid:2) S AB ∗ ( E A ∧ E B ) (cid:3) + S AB ∧ D ∗ ( E A ∧ E B )+ ∗ ( E A ∧ E C ) ∧ (Ω AB + Ω BA ) ∧ S BC . (27)Here, we have introduced a boundary term and reinstatedthe covariant derivative D with respect to the originalconnection Ω AB . On-shell, the latter is the Levi-Civitaconnection, which implies DE A = 0 and Q AB = 0, sothat integrating out the connection leads to S [ E ] = 12 κc (cid:90) M ∗ ( E A ∧ E C ) (28) ∧ (cid:104) (cid:101) R BC − S BD ∧ S DC (cid:105) . Up to a boundary term, this action is equivalent to theEinstein–Hilbert Lagrangian of GR, for any choice of theshift parameter S AB . GALILEAN CONNECTION AND ACTION
Just like how the Minkowski frame bundle metric η AB is associated to the Lorentzian tangent bundle metric g µν ,we can associate two GL ( n, R ) tensors t A , π AB , (29)satisfying t A π AB = 0, to the pre-non-relativistic (PNR)variables T µ and Π µν from Equation (2). Whereas theMinkowski metric η AB is fixed by the Lorentz subalge-bra of gl ( n, R ), the tensors (29) are left invariant by the Galilean algebra, which consists of spatial rotations andboosts with arbitrary velocity. We then introduce thedual tensors t A and π AB such that we have the orthogo-nality and completeness relations t A t A = − , t A π AB = t A π AB = 0 , (30a) δ AB = − t A t B + π AC π CB . (30b)We can introduce these Galilean tensors in a givenframe bundle, but they are not directly related to theLorentzian Palatini action (20) that we are working with.However, we can decompose the Minkowski metric as η AB = − t A t B + π AB , η AB = − t A t B + π AB , (31)following the ‘pre-non-relativistic’ decomposition (1) ofthe Lorentzian metric g µν , where we identify t A E Aµ = c T µ π AB Θ Aµ Θ Bµ = Π µν , (32a) t A Θ µA = − V µ /c, π AB Θ µA Θ νB = Π µν . (32b)Note that the frame tensors t A and π AB (and hence also T µ and Π µν ) are invariant under local Galilean transfor-mations but their duals t A and π AB (and hence also V µ and Π µν ) are not invariant.We now want to recast the Palatini action (20) in termsof the PNR Galilean variables defined above. Before ex-panding in powers of 1 /c , the action and the frames areinvariant under the Lorentz algebra, and this Galilean de-scription is somewhat unnatural. However, as outlined inthe Introduction, by rewriting the action for GR in thisway, we are preparing ourselves for a covariant descrip-tion of the non-relativistic Newton–Cartan geometry andits Galilei symmetry that appears in the expansion.Our main task now is to introduce a suitable Galileanconnection, corresponding to the frame version of the C ρµν connection (4) that was mentioned in the Introduction,and to identify its associated shift S AB . The main fea-tures of this connection are that it is compatible with theGalilean metric structure (29) and that it has, in somesense, the minimal required torsion.First, in analogy with the Lorentzian definition (13),the Galilean non-metricities are given by (G) Q A = − Dt A = t B Ω BA , (33a) (G) Q AB = Dπ AB / π AC Ω BC + π CB Ω AC ) / . (33b)It is useful to choose A = (0 , a ) so that we can write theGalilean tensors (29) as t A = δ A , π AB = δ Aa δ Bb δ ab , (34) E A = ( cT, E a ) , Θ A = ( − V /c, Θ a ) . (35)The non-zero components of the non-metricities (33) are (G) Q = Ω , (G) Q a = c − Ω a , (G) Q ab = Ω ( ab ) . (36)Here and in the following, we have introduced the ap-propriate factors of c so that the resulting objects suchas (G) Q a are O ( c ). Any connection for which all thesecomponents vanish is compatible with the Galilean PNRstructure, and will therefore lead to a Newton–Cartanmetric-compatible connection upon expansion.Note that we can raise and lower spatial indices usingthe spatial δ ab and δ ab . Together with (36), this allowsus to split a general gl ( n, R ) connection intoΩ [ ab ] , Ω a , (G) Q ab , (G) Q, (G) Q a , (37) in analogy with the Lorentzian decomposition (14). Thetorsion equations (8) then take the form T = cdT + c (G) Q a ∧ E a + c (G) Q ∧ T, (38) T a = d E a + Ω [ ab ] ∧ E b + c Ω a ∧ T + (G) Q ab ∧ E b . For Galilean connections with vanishing non-metricity,the time component of the torsion T = cdT is fixed in-dependent of the remaining connection components [24].This torsion is generically non-zero, and requiring that itvanishes means that the 1 /c expansion of the geometrydoes not accommodate time dilation.Instead, we will work with a Galilean metric-compatible connection which has T = cdT but vanishingspatial torsion. As is known from a tangent bundle per-spective [3], this does not fully fix the connection, andhere we use the equivalent of the connection (4),Ω [ ab ] = d E [ a (cid:0) Θ b ] , Θ c (cid:1) E c − d E [ a (Θ b ] , V ) T (39a) − d E c (Θ a , Θ b ) E c Ω a = c Ω a = 12 [ d E a (Θ c , V ) + d E c (Θ a , V )] E c . (39b)These connections can be related explicitly using the viel-bein postulate (11) and are connections for spatial rota-tions and Galilean boosts. To obtain the connection (39)as the solution (25) for ˜Ω AB , we need the shift S ab = c dT (Θ a , Θ b ) T, (40a) S a = c dT (Θ a , Θ c ) E c − cdT (Θ a , V ) T, (40b) S b = c dT (Θ b , Θ c ) E c − cdT (Θ b , V ) T (40c)+ 12 c [ d E b (Θ c , V ) + d E c (Θ b , V )] E c . With this, we can now work out the terms in thesecond-order action (28) explicitly in terms of Galilean-covariant objects. First, we have − η AB ∗ ( E A ∧ E C ) ∧ S BD ∧ S DC = δ ab δ cd c dT (Θ a , Θ c ) dT (Θ b , Θ d ) , (41)which is equivalent to the leading term in the La-grangian (5). The connection (39) is metric-compatible,which means that all non-metricities (36) vanish. As aresult, the components of its curvature (21) are given by R ab = d Ω ab + Ω ac ∧ Ω cb (42a) R a = cR a = d Ω a + Ω ab ∧ Ω b . (42b)From an algebraic perspective, R ab and R a are the cur-vatures corresponding to spatial rotations and Galileanboosts, see the Appendix. Their combination is invariantunder local Lorentz transformations before expanding,and after expanding it is invariant under local Galileanboosts at each order.With this, we can write the action with on-shell con-nection (28) in a Galilean-covariant way as S = 12 κ (cid:90) M (cid:20) c dT (Θ a , Θ b ) dT (Θ a , Θ b ) (43)+ R ab (Θ a , Θ b ) − c R a ( V, Θ a ) (cid:21) Ed n x. with E = det( T µ , E aµ ). This action is still Lorentz invari-ant, but it is constructed using Galilean building blocks.It precisely reproduces the PNR form of the Einstein–Hilbert Lagrangian (5). Given the shift (40), the equiv-alent decomposition of the off-shell action (27) in termsof a Galilean-compatible covariant derivative and non-metricity can then be used for the non-relativistic expan-sion of general relativity in the first-order formulation. DISCUSSION AND OUTLOOK
Through an appropriate field redefinition, we haveobtained an alternative formulation of the Palatini ac-tion, whose equations of motion give rise to a connectionthat is adapted to the Galilean-covariant Newton–Cartanstructure that arises in a non-relativistic 1 /c expan-sion of GR. By identifying the Lorentzian and Galileannon-metricity associated to a general affine connection,we also see how our field redefinition naturally leads tononzero torsion. The resulting action is equivalent toGR and allows us to extend its non-relativistic expan-sion to higher orders in a more tractable manner. Wealso expect that this action will be of use in computingconserved boundary charges, since this is typically moreaccessible in a first-order formulation. The 1 /c expan-sion of GR has been shown to reproduce the 1PN ap-proximation [7]. Verifying if this relation holds beyond1PN is cumbersome in the second-order formalism, andour first-order description is expected to ameliorate thissituation. Furthermore, since our formulation is off shell,it is also applicable to other astrophysical problems.Additionally, we can apply our procedure to the ultra-relativistic c → c gives a novel ap-proach to Carroll geometry (together with its sublead-ing corrections), which is connected to the geometry ofnull surfaces in GR [27] and is therefore potentially rel-evant to the dynamics of black hole horizons [28] andthe holographic description of asymptotically flat spaceat null infinity [29, 30]. This geometry is constructed us-ing the Carroll-invariant tensors t A and π AB , which haveopposite index structure compared to their Galilei coun- terparts. In this sense, the Carrollian notion of metric-compatibility is the dual of the Galilean case consideredabove, see also Ref. [31]. We will address both the non-relativistic and ultra-relativistic expansion of our novelaction and the corresponding geometry in more detailin upcoming work [32]. Different approaches to frameand/or first-order formulations of non-relativistic andultra-relativistic limits and expansions of gravity havepreviously been considered in Refs. [26, 33, 34].Finally, our field redefinition procedure can also beused to obtain Palatini-type actions for GR whose equa-tion of motion lead to a connection with vanishing Rie-mann tensor but non-trivial torsion or non-metricity.This corresponds to the teleparallel or symmetric telepar-allel formulation of GR, respectively, see also [35]. Acknowledgments.
We thank Stefano Baiguera,Jørgen Sandøe Musaeus and especially Dieter Van denBleeken and Manus Visser for useful discussions. Thework of DH is supported by the Swiss National ScienceFoundation through the NCCR SwissMAP. The workof JH is supported by the Royal Society UniversityResearch Fellowship “Non-Lorentzian Geometry inHolography” (grant number UF160197). The workof NO and GO is supported in part by the project“Towards a deeper understanding of black holes withnon-relativistic holography” of the Independent ResearchFund Denmark (grant number DFF-6108-00340) andthe Villum Foundation Experiment project 00023086.
APPENDIX: ALGEBRAIC PERSPECTIVE
We have considered Lorentzian and Galilean decom-positions of a general affine gl ( n, R ) connection based onthe corresponding notions of metric-compatibility in themain text. Alternatively, we can understand our resultsfrom a more algebraic perspective. One can obtain thegeometry associated to a generic frame bundle from agauge connection A that is associated to the affine group R n (cid:111) GL ( n, R ), which we can parametrize as A = E A P A + 12 Ω AB M AB . (44)Here, the translations P A and the matrices M AB generatethe affine algebra,[ M AB , M CD ] = − δ DA M CB + δ BC M AD (45a)[ M AB , P C ] = δ BC P A . (45b)Given a Minkowski metric η AB we can raise and lowerindices on the matrix generators M AB = η BC M AC andthe connection Ω AB = η BC Ω AC , so that we can splitthem in symmetric and antisymmetric components, S AB = M ( AB ) , J AB = M [ AB ] , (46a) A = E A P A + 12 Ω [ AB ] J AB + 12 Q AB S AB . (46b)The antisymmetric J AB generate the Lorentz algebraand couple to the metric-compatible connection Ω [ AB ] ,whereas the symmetric generators S AB couple to theLorentzian non-metricity Q AB = Ω ( AB ) . This corre-sponds to the decomposition in Equation (14).Alternatively, using A = (0 , a ) corresponding to Equa-tion (34), we can decompose the affine algebra as J ab = M [ ab ] , G a = M a , C a = M a ,H = P , P a , S ab = M ( ab ) , S = M . (47)Here, the translations are split in time translations H and space translations P a , and the spatial rotations J ab and Galilean boosts G a generate the Galilei subalgebraof gl ( n, R ). This induces the decomposition of the affineconnection (44) in the Galilean-compatible componentsΩ [ ab ] and Ω a = c Ω a as well as the non-metricities (G) Q ab = Ω ( ab ) , (G) Q a = c − Ω a , (G) Q = Ω , (48)corresponding to Equation (37)Finally, note that J ab and C a generate the Carrollsubalgebra of gl ( n, R ), which induces the correspondingCarrollian decomposition of Ω AB that was briefly men-tioned in the Conclusion. In particular, we see that theGalilean boost generator G a corresponds to a Carrolliannon-metricity component, while the Carrollian boost gen-erator C a corresponds to a Galilean non-metricity com-ponent. In this sense, the Carroll decomposition of a gen-eral affine connection is therefore the dual of the Galileandecomposition, see also Ref. [31]. [1] R. Andringa, E. Bergshoeff, S. Panda, and M. de Roo,Class. Quant. Grav. , 105011 (2011), arXiv:1011.1145[hep-th].[2] M. H. Christensen, J. Hartong, N. A. Obers, and B. Rol-lier, Phys. Rev. D89 , 061901 (2014), arXiv:1311.4794[hep-th].[3] J. Hartong and N. A. Obers, JHEP , 155 (2015),arXiv:1504.07461 [hep-th].[4] E. Cartan, Ann. ´Ec. Norm. Super. , 325 (1923).[5] E. Cartan, Ann. ´Ec. Norm. Super. , 1 (1924).[6] G. Dautcourt, Class. Quant. Grav. , A109 (1997),arXiv:gr-qc/9610036 [gr-qc].[7] W. Tichy and E. E. Flanagan, Phys. Rev. D84 , 044038(2011), arXiv:1101.0588 [gr-qc].[8] D. Van den Bleeken, Class. Quant. Grav. , 185004(2017), arXiv:1703.03459 [gr-qc].[9] D. Hansen, J. Hartong, and N. A. Obers, Phys. Rev.Lett. , 061106 (2019), arXiv:1807.04765 [hep-th]. [10] D. Hansen, J. Hartong, and N. A. Obers, JHEP , 145(2020), arXiv:2001.10277 [gr-qc].[11] D. T. Son, (2013), arXiv:1306.0638 [cond-mat.mes-hall].[12] K. Jensen, SciPost Phys. , 011 (2018), arXiv:1408.6855[hep-th].[13] J. Hartong, E. Kiritsis, and N. A. Obers, Phys. Lett. B746 , 318 (2015), arXiv:1409.1519 [hep-th].[14] M. Geracie, D. T. Son, C. Wu, and S.-F. Wu, Phys. Rev.
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