Newton-Cartan, Galileo-Maxwell and Kaluza-Klein
NNewton-Cartan, Galileo-Maxwelland Kaluza-Klein
Dieter Van den Bleeken and C¸ a˘gın Yunus
Physics Department, Bo˘gazi¸ci University34342 Bebek / Istanbul, TURKEY [email protected], [email protected]
Abstract
We study Kaluza-Klein reduction in Newton-Cartan gravity. In particularwe show that dimensional reduction and the nonrelativistic limit commute. Theresulting theory contains Galilean electromagnetism and a nonrelativistic scalar. Itprovides the first example of back-reacted couplings of scalar and vector matter toNewton-Cartan gravity. This back-reaction is interesting as it sources the spatialRicci curvature, providing an example where nonrelativistic gravity is more thanjust a Newtonian potential. a r X i v : . [ h e p - t h ] J un ontents B.1 Metric formulation of Newton-Cartan gravity . . . . . . . . . . . . 17B.2 Frame formulation of Newton-Cartan gravity . . . . . . . . . . . . . 19B.3 Galilean electromagnetism . . . . . . . . . . . . . . . . . . . . . . . 22B.4 Newton-Cartan-Maxwell dilaton theory . . . . . . . . . . . . . . . . 22
Newton-Cartan gravity was originally formulated to put Newtonian gravity in amanifestly coordinate invariant form [1]. In this formulation it is remarkably simi-lar to general relativity and indeed it was later shown that a careful nonrelativisticlimit of the Einstein equations leads directly to Newton-Cartan gravity [2–4]. Re-cently this classic subject has been of renewed interest due to its appearance in theholographic description of asymptotically non-AdS spaces [5–8] and applicationsin condensed matter physics [9, 10].In its most symmetric form the theory is formulated in terms of a one-form τ µ , a metric h µν and a connection ∇ (nc) µ , defined on a (1 , d ) dimensional manifold .These fields are however subject to a number of non-dynamical constraints andhence encode the real degrees of freedom of the theory in a somewhat convolutedway. By solving the constraints one can make the unconstrained fields manifest, be As in this paper various numbers of spatial dimensions are discussed we use the notation(1 , d ) where d indicates the number of spatial directions and the 1 represents the time direction.In the Lorentzian case the distinction between the two is made by the signature of the metric,in the Galilean case τ µ singles out the time direction.
1t at the cost of breaking manifest time reparameterization invariance. Althoughsuch a gauge-fixing would be a rather arbitrary thing to do in a relativistic the-ory, it is perfectly natural in a Galilean theory as all observers can agree on anabsolute time. One of the unconstrained fields making up Newton-Cartan gravityis the Newtonian gravitational potential Φ, but in addition there is also a spatialmetric h ij and a vector field C i . Although most authors are aware of these ad-ditional fields they typically tend to do away with them quickly, although thereare some exceptions [11, 12]. They seem to be justified in treating h ij and C i asless important/physical since at first sight the dynamical equations for these fieldsappear to be very rigid, at least in the case of the vacuum theory or in the presenceof a perfect fluid. The equation for the spatial metric is equivalent to vanishingRicci curvature, which in 3 spatial dimensions implies that it is locally flat. Thevectorfield in turn is forced to be harmonic which with some assumptions on theboundary conditions implies its curvature should vanish. An equal fate wouldawait the Newtonian potential, were it not that its Poisson equation is naturallysourced by mass density.It is clear then that the difference in standing between the fields is not so muchin the equations that govern them but rather in the (non-)appearance of sources.In the case of a perfect fluid this comes about because the pressure, which sourcesthe spatial components of the curvature in the relativistic theory, gets washedaway in the nonrelativistic limit and only the mass density remains. But onecould wonder if this is a general feature of all types of energy and momentum or ifthere are exceptions. This question is one of the main motivations for this work.Probably the main reason why coupling various types of matter and fields toNewton-Cartan gravity has not been investigated thoroughly (to the best of ourknowledge) is that the theory does not have a Lagrangian formulation. This makesit harder to investigate various possibilities consistent with the symmetries of thetheory or to decide which couplings are physical and minimal. On the other handit sounds somewhat strange that this would be of any concern since we know plentyof relativistic examples, can’t one simply take the nonrelativistic limit of those?The problem is that this limit is rather subtle and requires a starting ansatz. If oneknows the endpoint of the limit it is often easy to show how it arises, but withoutit it can be hard to come up with a good starting point. This will be illustrated inthis work as well, where we reproduce our results from a nonrelativistic limit, butthis limit includes a crucial subtlety (a choice of conformal frame) which wouldhave been hard to guess without knowing where we were heading. Here we are talking about fully back-reacted couplings, where the effect of these additionalfields on the gravitational sector is taken into account. There is a rather extensive literature[3, 13–27] on the simpler problem of studying various fields on a Newton-Cartan background,where the effect of these fields on that background is ignored.
GREMD NCNCMD c → ∞ c → ∞ KK KK
Figure 1: This commutative diagram shows the two ways one can go from generalrelativity (GR) to Newton-Cartan-Maxwell dilaton theory (NCMD). The first op-tion is to pass via Newton-Cartan gravity (NC) by first taking the nonrelativistic,i.e. c → ∞ , limit and then performing Kaluza-Klein reduction (KK). Or one couldgo via Einstein-Maxwell dilaton theory (EMD) by first performing Kaluza-Kleinreduction and then taking the nonrelativistic limit.Apart from the motivation just outlined before this work could be of somewhatmore applied interest as well. More elaborate versions of dimensional reductionplay an important role in string theory and supergravity in connecting the funda-mental UV theory of gravity to our 4 dimensional world. The nonrelativistic limitof these scenarios might lead to an approximation scheme that can introduce somesimplification in this rather complicated endeavor. Apart from its phenomeno-logical applications dimensional reduction also represents a powerful unifying andorganizing scheme. Instead of needing to study various theories in each dimen-sions separately it is often sufficient to study some high maximal dimension and3y reduction one then obtains a ’web’ of lower dimensional theories connected byvarious dualities. Supersymmetric theories are a good example of this and so itmight be that a nonrelativistic understanding of dimensional reduction can help inthe search for a supersymmetric extension of Newton-Cartan gravity [29–31] when d > Overview
The remainder of this paper is organized as follows. We start insection 2 by reviewing the basics of Newton-Cartan gravity. As the theory afterdimensional reduction will include nonrelativistic electromagnetism we shortly re-view this theory as well in section 3 and we put special emphasis on the presenceof magnetic charges as the explicit example we study includes those naturally.Section 4 contains one of the main results of this paper: our reduction ansatz andthe equations of motion of the nonrelativistic Kaluza-Klein theory. In section 5we perform the nonrelativistic limit of Einstein-Maxwell dilaton theory, a slightgeneralization of the Kaluza-Klein reduction of general relativity. We argue thatto perform this limit one has to go to a particular conformal frame and presentthe resulting nonrelativistic equations of motion. In the case one starts with theKaluza-Klein theory one obtains exactly the same result as the Kaluza-Klein re-duction of Newton-Cartan gravity presented in section 4, which can be seen as thesecond main result of the paper. To illustrate that this theory is physically inter-esting even when the number of spatial dimensions is 3 we present an example insection 6, namely that of a magnetic monopole, whose presence warps the spatialdimensions into a curved, albeit conformally flat, geometry.For the convenience of the reader we have also included two appendices. Inthe first, appendix A, we summarize some of our conventions and notation. Thesecond, appendix B, is written for those readers who are more familiar or prefer towork with the formulation of Newton-Cartan gravity that is invariant under the full(1 , d ) dimensional diffeomorphism group. We present all the theories discussed inthe paper; Newton-Cartan, Galileo-Maxwell and Newton-Cartan-Maxwell dilaton,in such a form and explain how the formulation in the main text can be obtainedby a particular partial gauge fixing that keeps invariance under time dependentspatial diffeomorphisms intact but breaks time reparametrization invariance.4
Newton-Cartan gravity
We start by shortly reviewing the structure of Newton-Cartan gravity in d spaceand 1 time dimensions, i.e. (1 , d ) dimensions, see e.g. [33–36] for further detailsand references. In this Galilean theory there is an absolute time coordinate that allobservers can agree on. We will make this explicit by working in a description ofthe theory where we have gauge-fixed time reparameterizations. We find it naturalto proceed this way and feel that it clarifies the role of the different physical fieldsin the theory, be it at the cost of losing some elegance. For convenience of readersmore familiar with the manifestly (1 , d ) diffeomorphism invariant formulation ofthe theory we have included appendix B where we show how the formulation usedhere is equivalent to these other, more common formulations.In the formulation here [37] the theory is manifestly invariant under time-dependent d -dimensional (spatial) diffeomorphisms generated by vectorfields ξ i ( t, x ).In addition there is a local u (1) gauge symmetry parametrized by λ ( t, x ). Newton-Cartan gravity contains the following three fieldsNewtonian potential: Φ( t, x ) δ Φ = L ξ Φ − C i ˙ ξ i − ˙ λ (1)Coriolis vector: C i ( t, x ) δC i = L ξ C i + h ij ˙ ξ j + ∂ i λ (2)Spatial metric: h ij ( t, x ) δh ij = L ξ h ij (3)Note that contrary to the time reparametrization invariant formulations in termsof constrained fields, the fields listed above are unconstrained.One sees that when the spatial diffeomorphisms are time-independent, i.e. ξ i ( x ), they act as usual purely through the spatial Lie derivative L ξ . When thespatial diffeomorphism is time dependent however there is an additional actionproportional to ˙ ξ i , mixing the different fields. In particular these time dependentdiffeomorphisms include local Galilean boosts, ξ i = v i t , and more generally can beinterpreted as describing a change to an arbitrary locally non-inertial frame.The dynamics of these fields is then described by the Newton-Cartan equationsof motion: R ij = 0 (4) −∇ j K ji = 2 h jk ∇ [ i ˙ h j ] k (5) −∇ i G i = 12 h ij ¨ h ij + 14 ˙ h ij ˙ h ij − K ij K ij + 4 π G N ρ (6)Here the covariant derivatives are with respect to the Levi-Civitta connection ofthe spatial metric h ij and R ij is the Ricci tensor for this connection. Furthermore We use coordinates x i , i = 1 . . . d on the spatial manifold. Furthermore we indicate timewith t and the time derivative with a dot. See appendix A for more details.
5e introduced the u (1)-invariant field strengths G i = − ∂ i Φ − ˙ C i δG i = L ξ G i + ( K ij − ˙ h ij ) ˙ ξ j − h ij ¨ ξ j (7) K ij = ∂ i C j − ∂ j C i δK ij = L ξ K ij − h k [ i ∇ j ] ˙ ξ k (8)We also included a source term in the form of ρ , the mass density, which couplesthrough Newton’s constant G N .Before we continue let us recall that the Newton-Cartan gravity formulatedabove can be obtained as a c → ∞ limit of the (1 , d ) dimensional Einstein equationsfor gravity coupled to a perfect fluid [37–39].Let us also point out to readers not familiar with this nonrelativistic gravitytheory that it has the simple solution h ij = δ ij , C i = 0 and ∂ i ∂ i Φ = 4 π G N ρ . Thislast equation is the well-known Poisson equation for the Newtonian gravitationalpotential. In case d = 3 this seems pretty much the most general solution as inthat case R ij = 0 implies that locally there exists a coordinate system such that h ij = δ ij . In the absence of non-trivial boundary conditions (5) then also implies C i = 0. This is no longer true in higher dimensions however and we’ll show insection 4 how this can be interpreted as the appearance of source terms in lowerdimensions. It will also be useful to review a few basics about Galilean electromagnetism.This theory was studied as a nonrelativistic limit of standard electromagnetismin [40] where it was formulated as a theory in flat space h ij = δ ij , C i = 0 thatis invariant under global Galilean transformations. Recently [41] investigated thefull conformal invariance of this theory. We will be interested in the extension toan arbitrary Newton-Cartan background, which was first performed in [3]. Someaspects of the inclusion of magnetic charge were discussed in [42]. For a cleardiscussion of the motivations and applications of nonrelativistic electromagnetismwe refer to [40].The fundamental fields of Galileo-Maxwell theory are an electric scalar poten-tial Ψ and a magnetic vector potential A i . They have a u (1) gauge-transformationand transform as a Galilean 1-form under time-dependent spatial diffeomorphisms:Electric potential: Ψ( t, x ) δ Ψ = L ξ Φ − A i ˙ ξ i − ˙ ζ (9)Vector potential: A i ( t, x ) δA i = L ξ A i + ∂ i ζ (10)It will be useful to introduce the u (1)-gauge invariant electric field and magnetic6urvature: E i = − ∂ i Ψ − ˙ A i δE i = L ξ E i + F ij ˙ ξ j (11) F ij = ∂ i A j − ∂ j A i δF ij = L ξ F ij (12)The Galilean version of Maxwell’s equations on an arbitrary Newton-Cartanbackground are then ∇ i E i = ρ (e) + 12 K ij F ij ∇ j F ji = − j i (e) (13) ∂ [ i F jk ] = ρ (m) ijk ∂ [ i E j ] = ˙ F ij + j (m) ij (14)Here the covariant derivatives are with respect to the Levi-Civitta connectionof h ij . Note however that apart from this ’minimal coupling’ of replacing partialderivatives with covariant ones, there is also a coupling to the Coriolis field strength K ij that sources the electric field. We like to point this out since although it isimplicitly there in the (1 , d ) dimensional covariant formulation of [3] it doesn’t getmuch attention in the literature. We’ll see one of its explicit effects in section 6.One can find the equivalent time-reparametrization invariant form of (13,14) inappendix B. In the formulation here this coupling is required by invariance underthe transformations (9).In the Galileo-Maxwell equations (13, 14) we have included both electric andmagnetic charge densities and currents as this will also be of some relevance inour example in section 6. We would like to point out the somewhat unusualconservation equations that follow from these nonrelativistic Maxwell equations: ∇ i j i (e) = 0 ˙ ρ (m) ijk + ∂ [ i j (m) jk ] = 0 (15)So although magnetic charge is locally conserved this is not the case for the electriccharge, which is only globally conserved, see [40] for further discussion.Most of the time we will leave the number of spatial dimensions d arbitrary andhence the magnetic charge is not a scalar density nor is its current a vector. When d = 3 one can easily connect back to the standard formulation via the definitions ρ (m) = √ h (cid:15) ijk ρ (m) ijk and j ( m ) ij = √ h(cid:15) ijk j ( m ) k . The conservation equation then takesthe familiar form ˙ ρ (m) + ∇ i j (m) i = 0.Finally let us point out that in [40], which is in d = 3 on flat space, it is stressedthat there are two ways to perform the nonrelativistic limit of electromagnetism,leading to what these authors call the electric respectively magnetic limit. If onehowever includes both magnetic and electric charges one sees that the magneticand electric limit lead to equivalent theories that are related by the simple , redefi-nitions E i → B i , B i → − E i together with ( ρ (e) , j i (e) ) → ( ρ ( m ) , j (m) i ) , ( ρ (m) , j (m) i ) → Admittedly when expressed in terms of the gauge potentials this field redefinition is non-local,which is a well known feature of electromagnetic duality. − ρ ( e ) , − j i (e) ), where B i = − (cid:15) ijk F jk . So we prefer to speak of the electric or mag-netic formulation of the nonrelativistic limit rather than use the terminology elec-tric vs magnetic limit. In this language the equations (13, 14) together with thedefinitions (11,12) constitute the magnetic formulation of Galilean electromag-netism. After having introduced some of the main nonrelativistic classical field theorieswe are now ready to come to the main point of the paper. In this section wewill start with Newton-Cartan gravity, as described in section 2, in (1 , d + 1)dimensions. We’ll consider field configurations that are independent of one spatialdirection and show that under this assumption the (1 , d + 1) dimensional theoryis equivalent to (1 , d ) dimensional Newton-Cartan gravity coupled to a scalar andelectromagnetic field. This is of course nothing but the nonrelativistic version ofthe classic Kaluza-Klein procedure, see [28] for a review.In this section we will denote the (1 , d +1) dimensional fields with a hat, and the(1 , d ) dimensional ones without. We will split the spatial coordinates as ( x i , y ), i = 1 , . . . , d where we assume all fields to be independent of y . Our reductionansatz is then ˆΦ( t, x, y ) = Φ( t, x ) −
12 Ω ( t, x )Ψ( t, x ) (16)ˆ C y ( t, x, y ) = − Ω ( t, x )Ψ( t, x ) (17)ˆ C i ( t, x, y ) = C i ( t, x ) − Ω ( t, x )Ψ( t, x ) A i ( t, x ) (18)ˆ h yy ( t, x, y ) = Ω ( t, x ) (19)ˆ h iy ( t, x, y ) = Ω ( t, x ) A i ( t, x ) (20)ˆ h ij ( t, x, y ) = Ω a ( t, x ) h ij ( t, x ) + Ω ( t, x ) A i ( t, x ) A j ( t, x ) (21)Naively one could assume the parameter a in the equation above to be any constantbut we will see immediately that the symmetries fix it to be zero. Indeed notonly this constant but the complete structure of the ansatz is fixed by demandingthat the lower dimensional fields transform properly under y -independent gaugetransformations. For gauge parameters λ ( t, x ) and ˆ ξ ˆ i ( t, x ) = ( ξ i ( t, x ) , ζ ( t, x )) thedecomposition of the fields presented above implies the presence of the following8ower dimensional fields and transformation lawsNewtonian potential: Φ( t, x ) δ Φ = L ξ Φ − C i ˙ ξ i − ˙ λ (22)Coriolis vector: C i ( t, x ) δC i = L ξ C i + Ω a h ij ˙ ξ j + ∂ i λ (23)Spatial metric: h ij ( t, x ) δh ij = L ξ h ij (24)KK potential: Ψ( t, x ) δ Ψ = L ξ Ψ − A i ˙ ξ i − ˙ ζ (25)KK vector: A i ( t, x ) δA i = L ξ A i + ∂ i ζ (26)Radion: Ω( t, x ) δ Ω = L ξ Ω (27)One observes now that the lower dimensional fields (Φ , C i , h ij ) transform properlyas the Newton-Cartan fields of section 2 only if we take a = 0 . (28)Once the ansatz has been fixed it is a matter of algebra to recast the (1 , d + 1)dimensional equations of motion as an equivalent set of d -dimensional equationsof motion. The result is ∇ i ∂ i Ω = 14 Ω F ij F ij ∇ i (cid:0) Ω E i (cid:1) = 12 Ω K ij F ij ∇ i (cid:0) Ω F ij (cid:1) = 0 (29)Ω R ij = ∇ i ∂ j Ω + 12 Ω F ik F jk ∇ j (cid:0) Ω K ji (cid:1) = − Ω ∂ i ( h jk ˙ h jk ) + ∇ j (Ω ˙ h ij ) − ∂ i ˙Ω + Ω F ij E j −∇ i (cid:0) Ω G i (cid:1) = 4 π G N ρ + 14 Ω (cid:16) ˙ h ij ˙ h ij − K ij K ij + 2 h ij ¨ h ij (cid:17) + 12 Ω E i E i + ¨ΩLet us point out that to obtain the last equation we also used that Newton’sconstant and the mass density get rescaled in the reduction: ˆG N ˆ ρ = Ω − G N ρ , withˆG N = 2 πR G N , where R is the radius of the internal dimension at spatial infinity.The equations above present a generalization of both Newton-Cartan gravityand Galilean electromagnetism. The first three equations represent a Galileanscalar and electromagnetic field on an arbitrary Newton-Cartan background, asreviewed in section 3. More interestingly we see that these fields also back-reactand appear as non-trivial sources in the right hand sides of the last three equa-tions describing the Newton-Cartan fields. A fully (1 , d ) diffeomorphism invariantversion of these equations is presented in appendix B for completion. Note that redefining ˜ C i = Ω a C i would cure the ξ transformation of h ij but would in turngive the wrong λ transformation.
9e will present an interesting non-trivial solution to the above theory in section6 but first we show how it can also be obtained as a nonrelativistic limit of ordinaryKaluza-Klein theory.
We start with a generic, relativistic Einstein-Maxwell dilaton theory in (1 , d ) space-time dimensions: L = (cid:112) − ˜ g (cid:18) ˜ R (gr) − ˜ g µν ∂ µ φ∂ ν φ − e − kφ ˜ H (cid:19) (30)Here ˜ R (gr) is the standard (general relativistic) Ricci scalar computed for theLorentzian metric ˜ g µν through its Levi-Cevitta connection and ˜ H = ˜ g µν ˜ g ρσ H µρ H νσ with H µν = ∂ µ B ν − ∂ ν B µ . The theories are parameterized by a constant k thatcan be freely chosen. The theory is equivalent to the Kaluza-Klein reduction of(1 , d + 1) dimensional Einstein gravity if one chooses k = (cid:115) d d −
1) (31)As we will argue below the nonrelativistic limit we perform is restricted to aparticular conformal frame. We start by performing a generic Weyl rescaling ofthe metric using the scalar φ , parameterized by a constant l :˜ g µν = e lφ g µν (32)Written in terms of the new metric the equations of motion following from (30)are equivalent to R (gr) µν = (cid:18)
12 + (1 − d ) l (cid:19) ∂ µ φ∂ ν φ + l ( d − ∇ µ ∂ ν φ + 12 g µν (cid:18) − kl + 12(1 − d ) (cid:19) e − k + l ) φ H (33)+ 12 e − k + l ) φ g ρσ H µρ H νσ g µν ∇ µ (cid:0) e − (2 k +(3 − d ) l ) φ H νρ (cid:1) = 0 (34) g µν ∇ µ ∂ ν φ = (1 − d ) l g µν ∂ µ φ∂ ν φ − k e − k + l ) φ H (35)10ere all covariant derivatives and the Ricci tensor R (gr) µν are with respect to theLevi-Cevitta connection of g µν , as is H = g µν g ρσ H µρ H νσ .We will now perform the nonrelativistic limit following Dautcourt [2, 3, 37–39],with the important subtlety that we apply his expansion in inverse powers of c tothe Lorentzian metric g µν which for l (cid:54) = 0 is not the Einstein-frame metric ˜ g µν thatappears in the standard form of the Lagrangian (30). We write η = c − and makethe following expansion ansatz for the various fields : g µν = α µν η − + γ µν + O ( η ) (36) g µν = h µν + β µν η + O ( η ) (37) φ = ϕ + O ( η ) (38) B µ = A µ + O ( η ) (39)It is assumed a priory that h µν is positive definite with a single zero eigenvalue.The fact that g µν is the inverse of g µν imposes then the constraints α µν = − τ µ τ ν h µν τ ν = 0 h µν γ νρ − β µν τ ν τ ρ = δ µρ (40)From this ansatz one can compute the curvatures and one finds that R (gr) µν = R ( − µν η − + R ( − µν η − + R (0) µν + O ( η ) (41) H µν = F µν + O ( η ) (42)Note that here R ( · ) µν are not the Ricci tensors of something, they are by definitionthe coefficients appearing in the expansion above. The same is true for F µν but itfollows from our expansion ansatz that F µν = ∂ µ A ν − ∂ ν A µ .We can now study the equations of motion (33-35) in this expansion. Tostart one observes that the right hand side of (33) has no term of order η − so itfollows that R ( − µν must vanish. A short computation shows that this condition isequivalent to h µν h ρσ ∂ [ µ τ ρ ] ∂ [ ν τ ,σ ] = 0, which is solved by τ µ = f ∂ µ t . If one demands,like in [2, 3], that the Levi-Civatta connection of g µν should remain non-singularin the η → f to be a constant , which in turncan be put to 1 by redefining t . This then has two consequences. For the strict c → ∞ limit it is sufficient to make an expansion in even powers of c . If onewants to compute the large c corrections to this limit one can take into account the odd powersas well. See [37,38] for more details. Let us also point out that we put the first subleading termsof φ and B µ to zero by hand. This will turn out to be a consistent choice, although one is notforced to do so. It is an interesting possibility to investigate the nonrelativistic limit without this constraint,something we plan to return to in the future. R ( − µν vanish. Comparing with the righthand side of (33), of which the term proportional to g µν is of order η − , we learnthat the limit is only consistent if we choose l = 12 k (1 − d ) (43)Via (32) this has the interpretation of fixing a particular conformal frame. Notethat the above choice doesn’t work for the case k = 0, which coincides with therather interesting case of Einstein-Maxwell theory.Secondly the choice f = 1 implies we can choose coordinates such that t = x and hence τ µ = δ µ . One can then solve the constraints (40) by β = 1 , β i = − h ij γ j , γ ij = h ij . (44)These considerations take care of all inverse powers of η and so the leadingterms in the equations of motion are of order O (1). One can now work out thesezeroth order equations of motion in terms of the coefficients in the expansions (36-39). To compare with our earlier discussions on Newton-Cartan gravity we rename A = − Ψ, γ i = C i , γ = −
2Φ and e − ϕ k = Ω. A straightforward calculation thenshows that these equations are equivalent to ∇ i ∂ i Ω = 14 Ω q +3 F ij F ij ∇ i (cid:0) Ω q +3 E i (cid:1) = 12 Ω q +3 K ij F ij ∇ i (cid:0) Ω q +3 F ij (cid:1) = 0 (45)Ω R ij = ∇ i ∂ j Ω + q Ω − ∂ i Ω ∂ j Ω + 12 Ω q +3 F ik F jk ∇ j (cid:0) Ω K ji (cid:1) = − Ω ∂ i ( h jk ˙ h jk ) + ∇ j (Ω ˙ h ij ) − ∂ i ˙Ω − q Ω − ˙Ω ∂ i Ω + Ω q +3 F ij E j −∇ i (cid:0) Ω G i (cid:1) = 14 Ω (cid:16) ˙ h ij ˙ h ij − K ij K ij + 2 h ij ¨ h ij (cid:17) + 12 Ω q +3 E i E i + ¨Ω + q Ω − ˙Ω where q = 2 k − dd − q = 0 and then the above equations coincide exactly with those presented in (29).This thus explicitly establishes the commutative diagram of figure 1.12n hindsight the fact that the reduction and the nonrelativistic limit commuteclarifies a few things. Starting with a relativistic Kaluza-Klein ansatz and expand-ing it in powers of η leads to the non-relativistic ansatz (19-21), if one chooses thecorrect conformal frame. In the limiting procedure the conformal frame gets fixedby the demand of a non-singular limit through (43). This is equivalent to the con-dition (28) in the nonrelativistic reduction which there followed from compatibilitywith the symmetries of the non-relativistic theory.Finally we would like to point out that our application of Dautcourt’s limitingprocedure vindicates its usefulness. It is a nonrelativistic limit that does not apriory start with an expansion around Minkowski space, in contrast to the morestandard post-Newtonian expansion [43]. When one considers only perfect fluidmatter, as Dautcourt did, the resulting nonrelativistic equations of motion forcethe zeroth order spatial metric to be flat. This then implies that the leading part ofthe metric is Minkowski after all, and the nonrelativistic limit then coincides withthe post-Newtonian expansion. This continues to hold for subleading orders [37].Here, where we also consider scalar and vector matter, there are nonrelativisticsolutions with a non-flat spatial metric, which implies the zeroth order metric is not Minkowski space and hence describes a limiting sector of Einstein-Maxwell-dilaton theory that is not captured by the post-Newtonian expansion. Indeed,translated back to the original relativistic Einstein-frame metric, the zeroth ordernonrelativistic solution corresponds to d ˜ s = Ω (cid:0) − ( c + 2Φ) dt + C i dx i dt + h ij dx i x j (cid:1) + O ( c − ) (47)while the post-Newtonian expansion [43] assumes metrics of the form d ˜ s = − ( c + 2Φ) dt + dx i dx i + O ( c − ) (48)So nonrelativistic solutions with non-trivial Ω, C i or h ij fall outside the standardpost-Newtonian expansion, see the next section for an explicit example. We end our paper with an explicit example, to show that the new nonrelativistictheory we presented is not a vacuous construction. We provide a solution of thereduced theory with a non-trivial source for the spatial curvature and discuss itsphysical interpretation as a magnetic monopole, with possible time dependentmagnetic charge.Both for simplicity and physical relevance we restrict ourselves here to d = 3.In this case the theory (29) is nothing but the dimensional reduction of pure13ewton-Cartan theory in (1 ,
4) dimensions so we know that the solution shouldoriginate from a Ricci flat 4d Euclidean metric with an isometry . These type ofmetrics are well-studied and have been fully classified in [45, 46]. They come intwo classes, the Killing vector can be of so called rotational or translational type.Here we will restrict ourselves to the second case which correspond to metrics alsoknown as the Gibbons-Hawking metrics [47]: d ˆ s = V dx i dx i + V − ( dy + ω i dx i ) (49)This metric is Ricci flat when ∂ i ω j − ∂ j ω i = (cid:15) ijk ∂ k V , which in particular impliesthat V is harmonic: ∂ i ∂ i V = 0.We can now simply read of fthe lower dimensional fields from the reductionansatz (19-21): h ij = V δ ij , Ω = V − / , A i = ω i ⇒ F ij = (cid:15) ijk ∂ k V (50)From this one can compute that ∇ i ∂ j V − / = 14 V − / (5 ∂ i V ∂ j V − δ ij ∂ k V ∂ k V − V ∂ i ∂ j V ) F ik F jk = V − ( δ ij ∂ k V ∂ k V − ∂ i V ∂ j V ) R ij = 14 V − (3 ∂ i V ∂ j V + δ ij ∂ k V ∂ k V − V ∂ i ∂ j V − V δ ij ∂ k ∂ k V )One then sees that the Ricci equation, the equation for Ω and that for F ij in (29)all become equivalent to ∂ i ∂ i V = 0 as expected.The remaining equations for K ij , E i and G i in (29) can then be written as: ∂ i E i = ( E i + K i ) V − ∂ i V (51) (cid:15) ijk ∂ j K k = − ∂ i ˙ V + (cid:15) ijk V − ∂ j V ( E k + K k ) (52) ∂ i G i = 12 V − (cid:0) K i K i − E i E i (cid:1) (53)where for convenience we defined K i = (cid:15) ijk K jk and took ρ = 0. These equationscan be solved by taking K i = − E i , G i = 0 and ∂ i E i = 0 (cid:15) ijk ∂ j E k = − ∂ i ˙ V (54) In [44] such Ricci flat metrics have been studied in the non-relativistic setting, but in aslightly different context. There they are form all of a 4d space-time, interpolating betweenrelativistic and non-relativistic. Here those manifolds form only the spatial part and directly ofa non-relativistic 5d geometry.
14s a last step let us make everything completely explicit in the simplest exam-ple, that of Euclidean Taub-NUT : V = 1 + pr , ( r = x i x i ) . (55)Note that p is a physical parameter of the solution that can not be absorbedin a coordinate transformation. In the lower dimensional theory this solutioncorresponds to a magnetic monopole, with scalar hair and a warped spatial metric: B i = 12 (cid:15) ijk F jk = − p x i r , h ij = (cid:16) pr (cid:17) δ ij Ω = (cid:114) rr + p . (56)In case p is a constant one can take E i = − K i = 0. We can make the solutionmore interesting however by choosing the charge to be a function of time: p ( t ). Inthat case ∂ i ˙ V = − ˙ pr r i and one then finds the following solutions for E i = − K i : E i = (cid:15) ijk ˙ p n j x k r ( r − x l n l ) (57)where n i as an arbitrary unit vector: n i n i = 1.Note that the electric field blows up along the semi-infinite line determinedby x i = rn i . This has the physical interpretation as the presence of a magneticcurrent that sources the electric field and is required to be there by the magneticcharge conservation equation (15).Finally we point out that this example provides a metric that is a zeroth ordersolution to the Einstein-Maxwell-dilaton theory (30) in a nonrelativistic expansionvia (47): d ˜ s = − c V − dt + V dx i dx i + O ( c − ) (58)Note that this type of solutions is not captured by the post-Newtonian expansion(48). Acknowledgements
We thank Eric Bergshoeff and Jan Rosseel for valuable discussions and commu-nication. DVdB is partially supported by TUBITAK grant 113F164 and by theBo˘gazi¸ci University Research Fund under grant number 13B03SUP7.
A Notation and conventions
In this appendix we collect a few notations and conventions. We assume y to be periodic, y (cid:39) y +2 πR , and the quantized magnetic charge is p (quant) = pR . pace and time We will denote with (1 , d ) the presence of d spatial and 1 timedirection for which we use coordinates x µ = ( t, x i ), i.e. µ = 0 , . . . d , i = 1 , . . . d and x = t . In the part of the paper where we perform the dimensional reduction wewill further split the spatial directions, in (1 , d +1) dimensions we write ˆ x ˆ i = ( x i , y )where i = 1 , . . . , d and ˆ i = 1 , . . . , d + 1.In most of the paper we work in a notation that is only covariant under d dimensional time dependent coordinate transformations, instead of under full (1 , d )dimensional coordinate transformations. As this is a language not commonly usedin the literature let us point out a few subtleties. Time derivative
We will denote the time derivative often with a dot. Ourconvention is that for a tensor ˙ T j ...j n i ...i m = ∂ t (cid:0) T j ...j n i ...i m (cid:1) (59)As the metric can be time dependent one cannot use the metric to raise or lowerindices on a tensor with a dot. For example: ˙ h ij means the time derivative of theinverse metric. This is not equal to the tensor obtained by raising the indices onthe time derivative of the metric:˙ h ij = − h ik h jl ˙ h kl (60) Lie derivative
We will denote the standard (1 , d )-dimensional Lie-derivative ofa (1 , d )-dimensional tensor by a (1 , d )-dimensional vectorfield ξ µ as £ ξ T ν ...ν n µ ...µ m = ξ λ ∂ λ T ν ...ν n µ ...µ m + T ν ...ν n λµ ...µ m ∂ µ ξ λ + . . . − T λν ...ν n µ ...µ m ∂ λ ξ ν − . . . (61)Most of the time we will however work with the d -dimensional, purely spatial, Lie-derivative. We define it with respect to a possibly time dependent d -dimensionalvector field ξ i acting on a possibly time dependent d -dimensional tensor as L ξ T j ...j n i ...i m = ξ k ∂ k T j ...j n i ...i m + T j ...j n ki ...i m ∂ i ξ k + . . . − T kj ...j n i ...i m ∂ k ξ j − . . . (62)In case we choose ξ µ = (0 , ξ i ) the two definitions are simply related, for examplein the case of a vector V µ = ( W, V i ) or a one-form ω µ = ( σ, ω i ): £ ξ W = L ξ W (63) £ ξ V i = L ξ V i − W ˙ ξ i (64) £ ξ σ = L ξ σ + ω i ˙ ξ i (65) £ ξ ω i = L ξ ω i (66)16 Time reparametrization invariant formulations
In this appendix we give a manifestly (1 , d ) dimensional diffeomorphism invariantform of all the equations of motion presented in the main text. We also explain howthe formulations there can be obtained from those in this appendix by a partialgauge fixing that leaves time dependent spatial diffeomorphisms manifest.
B.1 Metric formulation of Newton-Cartan gravity
We loosely follow [33, 34, 36] and refer to these texts for further details and refer-ences.
Setup
In the standard ’metric’ formulation of Newton-Cartan gravity one startswith the following fields: τ µ , h µν = h νµ , Γ (nc) ρµν = Γ (nc) ρνµ . (67)They transform as two tensors and a connection under (1 , d ) dimensional coordi-nate transformations δ ξ x µ = − ξ µ : δ ξ τ µ = £ ξ τ µ , δ ξ h µν = £ ξ h µν , δ ξ Γ (nc) ρµν = £ ξ Γ (nc) ρµν + ∂ µ ∂ ν ξ ρ (68)These fields are subject to the following covariant constraints: τ µ h µν = 0 ∇ (nc) µ τ ν = ∇ (nc) µ h νρ = 0 (69)Note that by the symmetry of the connection these constraints furthermore implythat ∂ [ µ τ ν ] = 0.The equations of motion are R (nc) µν = 4 π G N ρ τ µ τ ν h λ [ µ R (nc) ν ]( ρσ ) λ = 0 (70)where the above are the Ricci and Riemann tensor for the connection ∇ (nc) µ , ρ isthe mass density and G N is Newton’s constant. Exhibiting the degrees of freedom
The constraints in (69) involving thecovariant derivative reduce the degrees of freedom contained in the connection, butdo so differently than in the familiar case of general relativity. In the relativisticcase, where g µν is invertible, the constraint of compatibility between the metricand the connection leads to a unique expression of the connection in terms ofthe metric and hence the connection contains no additional degrees of freedom.17ere the situation is different, but one can continue analogously by solving for thedependence of the connection on τ µ and h µν explicitly. To do so one introducestwo new tensor fields: τ µ and h µν = h νµ : δ ξ τ µ = £ ξ τ µ δ ξ h µν = £ ξ h µν (71)These are defined so as to satisfy their own set of constraints: h µν h νρ + τ µ τ ρ = δ µρ τ ρ τ σ h ρσ = 0 (72)Note that these imply τ µ τ µ = 1 and h µν τ ν = 0. It is important to realize that theseconstraints don’t fix the new fields completely in terms of h µν and τ µ . This followsfrom the observation that the constraints (72) are invariant under the followinglocal symmetry [13]: δ χ h µν = δ χ τ µ = 0 , δ χ τ µ = − h µν χ ν , δ χ h µν = τ µ χ ν + τ ν χ µ − τ µ τ ν τ ρ χ ρ (73)To avoid introducing new degrees of freedom we are thus forced to consider theabove transformation as a gauge symmetry. The introduction of the new fields h µν and τ µ is very convenient however as they allow to solve the second constraint in(69) explicitly:Γ (nc) λµν = τ λ ∂ µ τ ν + 12 h λρ ( ∂ µ h νρ + ∂ ν h µρ − ∂ ρ h µν ) − h λρ K ρ ( µ τ ν ) (74)This solution is not uniquely determined in terms of h µν and τ µ as it includes anarbitrary two-form K µν = − K νµ . This two-form contains those degrees of freedomof Γ (nc) that are indepenent of h µν and τ µ . As the definition of K µν depends on achoice of h µν and τ µ it changes under the symmetry (73): δ χ Γ (nc) λµν = 0 δ χ K µν = − ∂ µ χ ν + ∂ ν χ µ + τ µ ∂ ν ( τ ρ χ ρ ) − τ ν ∂ µ ( τ ρ χ ρ ) (75)Note that although K µν is part of a connection it transforms as a tensor undercoordinate transformations: δ ξ K µν = £ ξ K µν (76) Gauge-fixing
Restricting ourselves to a local patch for simplicity the constraint ∂ [ µ τ ν ] = 0 implies we can write τ µ = ∂ µ t ( x ). Using the (1 , d ) dimensional diffeo-morphism invariance we can then choose τ µ = δ µ , i.e t = x , and we accordinglysplit the index µ = (0 , i ). By the constraints this gauge choice puts h µ = 0 τ = 1 h ij h jk = δ ik τ i = − h ij h j h = h i h j h ij (77)18urthermore this gauge-fixes the (1 , d ) dimensional diffeomorphisms to time depen-dent d dimensional diffeomorphisms (and constant time shifts): ξ µ = ( ξ , ξ i ( t, x )).We can simplify further by also using the χ transformation (73). Since δ χ h i = χ i we can always put h i = 0. The constraints then furthermore imply that τ i = 0and h i = 0. This second choice of gauge is preserved by the combination ofdiffeomorphisms and χ transformations that satisfy the relation χ i = − h ij ˙ ξ j (78)The remaining fields, where for convenience we define G i = K i , then transformunder these transformations as δh ij = L ξ h ij (79) δG i = L ξ G i + ( K ij − ˙ h ij ) ˙ ξ j − h ij ¨ ξ j (80) δK ij = L ξ K ij + ∂ i ( h jk ˙ ξ k ) − ∂ j ( h ik ˙ ξ k ) (81)One can now work out the equations of motion (70) in this gauge. As a firststep one computes that the connection decomposes asΓ (nc)0 µν = 0 Γ (nc) i = − G i Γ (nc) i j = 12 h ik ( ˙ h kj − K kj ) Γ (nc) kij = Γ kij (82)Some additional computation then reveals that the Trautman condition, i.e thesecond of these equations, is equivalent to ∇ [0 K ij ] = 0 ∇ [ i K jk ] = 0 (83)This can be explicitly solved by introducing two new fields: Φ and C i and writing G i = K i = −∇ i Φ − ˙ C i K ij = ∂ i C j − ∂ j C i (84)Note that these fields are only defined up to a new gauge transformation: δ λ Φ = − ˙ λ δ λ C i = ∂ i λ (85)The other transformations (80-81) furthermore imply that Φ and C i transformexactly as in (1-2).Finally a straightforward although somewhat cumbersome application of alge-bra shows that indeed the first equation of (70) is equivalent to (4-6). B.2 Frame formulation of Newton-Cartan gravity
In this subsection we mainly follow [36] to which we refer for further details andreferences. 19 etup
In this case one starts with the fields τ µ , e aµ , ω µab , (cid:36) µa , C µ , they have thefollowing local gauge transformations: δτ µ = ∂ µ ξ (86) δe aµ = ∂ µ ξ a + Λ ab e bµ − ξ b ω µab − ξ(cid:36) µa + χ a τ µ (87) δω µab = ∂ µ Λ ab + 2 (cid:36) µ [ ac Λ b ] c (88) δ(cid:36) µa = ∂ µ χ a + Λ ab (cid:36) µb − χ b ω µab (89) δC µ = ∂ µ λ − χ b e bµ + ξ b (cid:36) µb (90)These transformations generate the so called Bargmann algebra [48], a centralextension of the Gallilei algebra. For each field one can construct a gauge-covariantcurvature in the standard way: R ( τ ) µν = ∂ µ τ ν − ∂ ν τ µ (91) R ( e ) µν a = ∂ µ e aν − ω µab e bν + e bµ ω ν ab + τ µ (cid:36) ν a − (cid:36) µa τ ν (92) R ( ω ) µν ab = ∂ µ ω ν ab − ∂ ν ω µab − ω µc [ a ω ν b ] c (93) R ( (cid:36) ) µν a = ∂ µ (cid:36) ν a − ∂ ν (cid:36) µa − ω µab (cid:36) ν b + (cid:36) µb ω ν ab (94) R ( C ) µν = ∂ µ C ν − ∂ ν C µ + (cid:36) µb e bν − e bµ (cid:36) ν b (95)As a first step one reduces the number of independent fields by imposing threecurvature constraints: R ( τ ) µν = 0 R ( e ) µν a = 0 R ( C ) µν = 0 (96)The two last ones allow to solve for ω and (cid:36) in terms of the other fields: ω µab = ∂ [ µ e aν ] e νb + ∂ [ ρ e µ ] b e ρa − ∂ [ ν e cρ ] e µc e ρa e νb + τ µ e ρa e νb ∂ [ ρ C ν ] (97) (cid:36) µa = τ µ e νa τ ρ ∂ [ ν C ρ ] + e νa ∂ [ ν C µ ] + e ρa e µb τ ν ∂ [ ρ e bν ] + τ ρ ∂ [ µ e aρ ] (98)Note that this solution makes use of two additional fields e µa and τ µ . These carryhowever no new independent degrees of freedom as they are uniquely definedthrough the relations e aµ e µb = δ ab , τ µ τ µ = 1 , e aµ τ µ = 0 , τ µ e µa = 0 (99)Note that the expression (97) also defines a connection Γ (nc) ρµν on tangent spaceby demanding the frame to be parallel with respect to a mixed covariant derivative: ∇ µ e aν = ∂ µ e aν − Γ (nc) ρµν e aρ − ω µab e ν b − (cid:36) µa τ ν = 0 (100)20his connection is exactly the same as (74) under the identifications h µν = e µa e νb δ ab , h µν = e aµ e bν δ ab K µν = ∂ µ C ν − ∂ ν C µ (101)Finally, in this formulation the dynamics is given by the following equations: e νa R ( (cid:36) ) µν a = 4 π G N ρ τ µ e µa e νb R ( ω ) µν ab = 0 (102) Gauge fixing
The first of the constraints (96) is ∂ [ µ τ ν ] = 0 so we can locallyalways choose τ µ = ∂ µ t . We can then use the gauge transformation (86) to put τ µ into the form τ µ = δ µ . Note that this implies e a = 0. This choice of gauge hasfixed the ξ gauge transformations up to ξ =cst. Now note that we can use the χ a gauge transformation, corresponding to a local translation, to put the e a = 0. Thisimplies then that τ µ = δ µ and and furthermore e ai e ja = δ ji , e ai e ib = δ ba . It furtherfixes the χ a transformations up to those that can be undone by a compensatingcombination of local translations, rotations and constant time shifts. Workingin terms of spatial diffeomorphisms instead of local translations by introducing ξ a = e ai ξ i the gauge choice we made is invariant under transformations satisfying λ a = − e ai ˙ ξ i + ω ia ξ i + ζ(cid:36) a (103)Plugging this into the gauge transformations of the remaining fields, where wewrite C = − Φ one finds after some algebra that: δe ai = L ξ e ai + Λ ab e bi (104) δ Φ = − ˙ λ + L ξ Φ − C i ˙ ξ i (105) δC i = ∂ i λ + h ij ˙ ξ j + L ξ C i (106)These are exactly (1-3) via (101).The dynamical equations (102) become in this gauge R ( (cid:36) )0 aa = 4 πGρ (107) R ( (cid:36) ) iaa = 0 (108) R ( ω ) iaab = 0 (109)These are seen to be equivalent to (4-6) after working through some algebra. The equation R ( ω )0 aab = 0 follows automatically from the others by a Bianchi identity onthe curvatures. .3 Galilean electromagnetism Just as in the relativistic case the theory can be expressed in terms of a one-formpotential A µ that transforms as δA µ = £ ξ A µ + ∂ µ ζ (110)The gauge invariant fieldstrength is as usual F µν = ∂ µ A ν − ∂ ν A µ . The equationsof motion on an arbitrary Newton-Cartan background are then h µν ∇ (nc) µ F νρ = j (e) ρ ∂ [ µ F νρ ] = j (m) µνρ (111)It takes only a short calculation to show that by the gauge fixing of appendix B.1the above equations and transformation rules coincide with (13,14) and (9,10),when one makes the identifications A = − Ψ, ρ (e) = j (e)0 and ρ (m) ij = j (m) ij . B.4 Newton-Cartan-Maxwell dilaton theory
In section 5 we derived a family of nonrelativistic theories, parameterized by acoupling constant q , that generalizes (1 , d )-dimensional Newton-Cartan gravity toinclude scalar and vector fields. When q = 0 this theory is the Kaluza-Kleinreduction of (1 , d + 1)-dimensional Newton-Cartan gravity. The theory containsin addition to the Newton-Cartan fields (67) a vector field (110) and a scalar field: δ Ω = £ ξ Ω. Its manifest (1 , d ) diffeomorphism invariant equations of motion aregiven by h µν ∇ (nc) µ ∂ ν Ω = 14 Ω q +3 h µν h ρσ F µρ F νσ (112) h µν ∇ (nc) µ (cid:0) Ω q +3 F νρ (cid:1) = 0 (113)Ω R (nc) µν = ∇ (nc) µ ∂ ν Ω + q Ω − ∂ µ Ω ∂ ν Ω + 12 Ω q +3 h ρσ F µρ F νσ (114) h λ [ µ R (nc) ν ]( ρσ ) λ = 0 (115)After the same gauge fixing procedure as in section B.1 these equations can beshown to be identical to (45). References [1] E. Cartan, “Sur les vari´et´es a connexion affine et la th´eorie de la relativit´eg´en´eralis´e.,”
Annales Sci. Ecole Norm. Sup. (1923) 325–412.222] G. Dautcourt, “Die Newtonske Gravitationstheorie als Strenger Grenzfallder Allgemeinen Relativit¨atheorie,” Acta Phys. Pol. (1964) 637.[3] H. K¨unzle, “Covariant Newtonian limit of Lorentz space-times,” GeneralRelativity and Gravitation (1976) 445–457.[4] J. Ehlers, “Uber den Newtonschen Grenzwert der EinsteinschenGravitationstheorie,” Grundlagenprobleme der modernen Physik ed J Nitschet al (1981) .[5] M. H. Christensen, J. Hartong, N. A. Obers, and B. Rollier, “TorsionalNewton-Cartan Geometry and Lifshitz Holography,”
Phys. Rev.
D89 (2014)061901, arXiv:1311.4794 [hep-th] .[6] M. H. Christensen, J. Hartong, N. A. Obers, and B. Rollier, “BoundaryStress-Energy Tensor and Newton-Cartan Geometry in LifshitzHolography,”
JHEP (2014) 057, arXiv:1311.6471 [hep-th] .[7] J. Hartong, E. Kiritsis, and N. A. Obers, “Lifshitz space–times forSchr¨odinger holography,” Phys. Lett.
B746 (2015) 318–324, arXiv:1409.1519 [hep-th] .[8] J. Hartong, E. Kiritsis, and N. A. Obers, “Schr¨odinger Invariance fromLifshitz Isometries in Holography and Field Theory,”
Phys. Rev.
D92 (2015) 066003, arXiv:1409.1522 [hep-th] .[9] D. T. Son, “Newton-Cartan Geometry and the Quantum Hall Effect,” arXiv:1306.0638 [cond-mat.mes-hall] .[10] M. Geracie, D. T. Son, C. Wu, and S.-F. Wu, “Spacetime Symmetries of theQuantum Hall Effect,”
Phys. Rev.
D91 (2015) 045030, arXiv:1407.1252[cond-mat.mes-hall] .[11] G. Dautcourt, “COSMOLOGICAL CORIOLIS FIELDS IN THENEWTON-CARTAN THEORY,”
Gen. Rel. and Grav. (1990) 765–769.[12] J. Ehlers, “Examples of Newtonian limits of relativistic spacetimes,” Classical and Quantum Gravity (1997) 119–126.[13] C. Duval and H. P. Kunzle, “Minimal Gravitational Coupling in theNewtonian Theory and the Covariant Schrodinger Equation,” Gen. Rel.Grav. (1984) 333. 2314] H. P. Kunzle and C. Duval, “DIRAC FIELD ON NEWTONIANSPACE-TIME,” Annales Poincare Phys. Theor. (1984) 363–384.[15] M. de Montigny, F. C. Khanna, and A. E. Santana, “Nonrelativistic waveequations with gauge fields,” Int. J. Theor. Phys. (2003) 649–671.[16] E. S. Santos, M. de Montigny, F. C. Khanna, and A. E. Santana, “Galileancovariant Lagrangian models,” J. Phys.
A37 (2004) 9771–9789.[17] D. T. Son and M. Wingate, “General coordinate invariance and conformalinvariance in nonrelativistic physics: Unitary Fermi gas,”
Annals Phys. (2006) 197–224, arXiv:cond-mat/0509786 [cond-mat] .[18] K. Jensen, “On the coupling of Galilean-invariant field theories to curvedspacetime,” arXiv:1408.6855 [hep-th] .[19] K. Jensen, “Aspects of hot Galilean field theory,”
JHEP (2015) 123, arXiv:1411.7024 [hep-th] .[20] M. Geracie, K. Prabhu, and M. M. Roberts, “Curved non-relativisticspacetimes, Newtonian gravitation and massive matter,” J. Math. Phys. no. 10, (2015) 103505, arXiv:1503.02682 [hep-th] .[21] M. Geracie, K. Prabhu, and M. M. Roberts, “Fields and fluids on curvednon-relativistic spacetimes,” JHEP (2015) 042, arXiv:1503.02680[hep-th] .[22] J. F. Fuini, A. Karch, and C. F. Uhlemann, “Spinor fields in generalNewton-Cartan backgrounds,” arXiv:1510.03852 [hep-th] .[23] R. Auzzi, S. Baiguera, and G. Nardelli, “On Newton-Cartan traceanomalies,” arXiv:1511.08150 [hep-th] .[24] R. Banerjee, A. Mitra, and P. Mukherjee, “A new formulation ofnon-relativistic diffeomorphism invariance,” Phys. Lett.
B737 (2014)369–373, arXiv:1404.4491 [gr-qc] .[25] R. Banerjee, A. Mitra, and P. Mukherjee, “Localization of the Galileansymmetry and dynamical realization of Newton-Cartan geometry,”
Class.Quant. Grav. no. 4, (2015) 045010, arXiv:1407.3617 [hep-th] .[26] R. Banerjee, A. Mitra, and P. Mukherjee, “General algorithm fornonrelativistic diffeomorphism invariance,” Phys. Rev.
D91 no. 8, (2015)084021, arXiv:1501.05468 [gr-qc] .2427] R. Banerjee and P. Mukherjee, “New approach to nonrelativisticdiffeomorphism invariance and its applications,” arXiv:1509.05622[gr-qc] .[28] D. Bailin and A. Love, “KALUZA-KLEIN THEORIES,”
Rept. Prog. Phys. (1987) 1087–1170.[29] R. Andringa, E. A. Bergshoeff, J. Rosseel, and E. Sezgin, “3DNewton–Cartan supergravity,” Class. Quant. Grav. (2013) 205005, arXiv:1305.6737 [hep-th] .[30] E. Bergshoeff, J. Rosseel, and T. Zojer, “Newton–Cartan (super)gravity as anon-relativistic limit,” Class. Quant. Grav. no. 20, (2015) 205003, arXiv:1505.02095 [hep-th] .[31] E. Bergshoeff, J. Rosseel, and T. Zojer, “Newton-Cartan supergravity withtorsion and Schr¨odinger supergravity,” JHEP (2015) 180, arXiv:1509.04527 [hep-th] .[32] B. Julia and H. Nicolai, “Null Killing vector dimensional reduction andGalilean geometrodynamics,” Nucl. Phys.
B439 (1995) 291–326, arXiv:hep-th/9412002 [hep-th] .[33] P. Havas, “Four-Dimensional Formulations of Newtonian Mechanics andTheir Relation to the Special and the General Theory of Relativity,”
Rev.Mod. Phys. (1964) 938–965.[34] H. K¨unzle, “Galilei and Lorentz Structures on Space-Time: Comparison ofthe Corresponding Geometry and Physics,” Ann. Inst. Henry Poincar´e (1972) 337.[35] C. Duval and P. A. Horvathy, “Non-relativistic conformal symmetries andNewton-Cartan structures,” J. Phys.
A42 (2009) 465206, arXiv:0904.0531[math-ph] .[36] R. Andringa, E. Bergshoeff, S. Panda, and M. de Roo, “Newtonian Gravityand the Bargmann Algebra,”
Class. Quant. Grav. (2011) 105011, arXiv:1011.1145 [hep-th] .[37] G. Dautcourt, “PostNewtonian extension of the Newton-Cartan theory,” Class. Quant. Grav. (1997) A109–A118, arXiv:gr-qc/9610036 [gr-qc] .2538] G. Dautcourt, “ON THE NEWTONIAN LIMIT OFGENERAL-RELATIVITY,” ACTA PHYSICA POLONICA B (1990)755–765.[39] W. Tichy and E. E. Flanagan, “Covariant formulation of thepost-1-Newtonian approximation to General Relativity,” Phys. Rev.
D84 (2011) 044038, arXiv:1101.0588 [gr-qc] .[40] M. L. Bellac and J. M. L´evy-Leblond, “Galilean electromagnetism,”
NuovoCimento B (1973) 217–233.[41] A. Bagchi, R. Basu, and A. Mehra, “Galilean Conformal Electrodynamics,” JHEP (2014) 061, arXiv:1408.0810 [hep-th] .[42] V. I. Strazhev, “Galilean Invariance and Magnetic Charge,” InternationalJournal of Theoretical Physics (1974) .[43] E. Poisson and C. Will, Gravity: Newtonian, Post-Newtonian, Relativisticstring . Cambridge University Press, 2014.[44] M. Dunajski and J. Gundry, “Non-relativistic twistor theory andNewton–Cartan geometry,” arXiv:1502.03034 [hep-th] .[45] C. P. Boyer and J. D. Finley, III, “Killing Vectors in Selfdual, EuclideanEinstein Spaces,”
J. Math. Phys. (1982) 1126.[46] J. Gegenberg and A. Das, “Stationary Riemannian Space-Times withSelf-Dual Curvature,” Gen. Rel. Grav. (1984) .[47] G. W. Gibbons and S. W. Hawking, “Gravitational Multi - Instantons,” Phys. Lett.
B78 (1978) 430.[48] C. Duval, G. Burdet, H. P. Kunzle, and M. Perrin, “Bargmann Structuresand Newton-cartan Theory,”
Phys. Rev.