The Teleparallel Equivalent of Newton-Cartan Gravity

James Read⇤ Hertford College, University of Oxford Oxford OX2 6GG, United Kingdom

Nicholas J. Teh† University of Notre Dame Notre Dame, IN 46556, USA (Dated: July 31, 2018) We construct a notion of teleparallelization for Newton-Cartan theory, and show that the telepar- allel equivalent of this theory is Newtonian gravity; furthermore, we show that this result is consistent with teleparallelization in general relativity, and can be obtained by null-reducing the teleparallel equivalent of a five-dimensional gravitational wave solution. This work thus strengthens substan- tially the connections between four theories: Newton-Cartan theory, Newtonian gravitation theory, general relativity, and teleparallel gravity.

Newton-Cartan theory and the teleparallel analogy— tion. This thus raises the question of why such an anal- Although Newton-Cartan theory (NCT) was originally ogy exists, and whether one can use geometric tools to conceived as a ‘geometrized’ version of Newtonian grav- provide a satisfactory explanation of these relationships. itation [12], this theory has recently been found to have Indeed, on the basis of the analogy, one might suspect a wide and impressive range of further applications: it that standard Newtonian gravity is a kind of teleparallel provides an improved model of the fractional quantum theory, and—even more tantalizingly—that dimensional Hall e↵ect [24, 43], a geometric foundation for Horava- reduction of the kind described above ‘commutes’ with Lifschitz gravity [27], and a framework for non-relativistic this teleparallelization, in the sense that if we dimen- holography [13, 14], to name just a few examples. Many sionally reduce the teleparallel form of the Bargmann- of these applications owe their success to two power- Eisenhart solution, we precisely recover standard Newto- ful techniques for analyzing NCT. First, the recently- nian gravity. This set of conjectural relationships can be developed vielbein formalism for NCT [4, 23] has played illustrated schematically as follows: an indispensable role in understanding the hidden local teleparallelization Galilean invariance of NCT—akin to the hidden local GR GR TPG Lorentz invariance of general relativity (GR)—and in an- alyzing the coupling of matter fields to a general Newton- null reduce null reduce? Cartan background spacetime. Second, the technique NCT NG of null dimensional reduction [17, 30]—which allows one teleparallelization ? to formulate a D-dimensional solution of NCT as a cer- NC tain (D + 1)-dimensional gravitational wave (Bargmann- As will be discussed further below, teleparallelizationGR Eisenhart) solution to GR—has provided an especially is standardly defined in the TPG literature as the ‘gaug- ecient method of using a relativistic spacetime to an- ing’ of translations within the relativistic (Poincar´e) alyze the symmetries [18] and dynamics [19] of a large gauge group while setting the spin connection to be pure class of non-relativistic mechanical systems. gauge; its non-relativistic analog teleparallelizationNC Despite the obvious relevance and power of these tech- will be defined below. niques, they have not yet been applied to what is in This letter answers the above question by combining some sense the fundamental theorem of NCT, viz., the vielbein and dimensional reduction techniques to shed Trautman Recovery Theorem [45], which asserts that new light on the Trautman Recovery Theorem, thereby curved Newton-Cartan gravity is ‘empirically equiva- showing that (i) Newtonian gravity is the teleparallel lent’ to the standard (flat, but with forces) Newtonian equivalent of Newton-Cartan theory (i.e. by computing theory of gravitation. Furthermore, it seems to have the bottom edge of the diagram), and (ii) that teleparal- gone unnoticed in the literature that such a statement lelization commutes with dimensional reduction (i.e. by is highly analogous to the key idea driving teleparallel computing the top and right edges of the diagram). gravity (TPG), which is that generically curved models While the relevance of the dimensional reduction tech- of GR are empirically equivalent (modulo subtleties—see nique to this result is obvious, it will perhaps help to e.g. [41, 42]) to a theory with a flat but torsionful connec- highlight why the vielbein formulation of NCT plays such an important role in our argument. In the modern theory of TPG [42], the vielbein apparatus is crucial for ‘gaug- ing the translations’ and thereby characterizing torsion ⇤ [email protected] as the field strength of this gauge field. Analogously, and † [email protected] as we will see below, if we want to teleparallelize NCT 2 then we must first understand what the gauge group of an extended vielbein [23] for NCT theory is a 1-form field I a NCT is from the perspective of its vielbein formulation— eµ =(⌧µ,eµ ,mµ), which is valued in the extended in- surprisingly, it turns out that ‘gauging the translations’ 3 ternal tangent space R R u (1)mass (i.e., the direct sum in this context leads to the concept of a ‘mass torsion’ of infinitesimal time translations, spatial translations, that plays the role of force in Newtonian gravity. and mass translations). Following [23], we equip the ex- We proceed by first reviewing vielbein NCT and TPG tended internal tangent space with a Minkowski metric for the reader. This done, we apply the teleparallelization ⌘IJ, and use light-cone coordinates in which ⌘ab = ab techniques to vielbein NCT, demonstrating that the re- and ⌘ + = ⌘+ = 1. As extensively discussed in [23], sulting theory is Newtonian gravitation theory, and that this vielbein is ‘extended’ in the sense that it includes a notion of mass torsion is associated with force in this the U(1)mass gauge field mµ, which is needed to correctly theory. This establishes the bottom leg of the above dia- couple a massive point particle (and matter fields) to a gram. Finally, we construct the teleparallel equivalent of Newton-Cartan background. We also note that a (non- the Bargmann-Eisenhart solution of GR and show that it µ µ µ extended) frame e A =(v ,e a) can be defined in the can be null reduced to obtain standard Newtonian grav- µ A µ A µ A usual way by means of e Ae = and eµ e B = B . itation, thereby completing the above diagram. The associated (extended) spin connection 1-form Review of vielbein NCT —In this paper, we follow the I !µJ is valued in gal0 and consists of the boost connec- index conventions of [7, 23]—that is, for spacetime in- a a a tion !µband the rotation connection !µ := !µ 0 (be- dices we use M,N,... running from 0 to 4; µ, ,... run- low, we will discuss how the spin connection is related to ning from 0 to 3; and the Latin index v for a privileged I a familiar spacetime connection). We note that eµ and spacetime null direction. On the other hand, for (inter- ! I transform under local Galilean boosts of the form nal) tangent space indices we use I,J,... running from 0 µJ vµ vµ + kbe µ. to 4; a, b, . . . running from 1 to 3; A, B, . . . running from 7! b 0 to 3; and for tangent space null directions (when The above objects constitute the Cartan data for lo- in light-cone± coordinates). We adopt a boldface nota- cally defining a Cartan gauge theory (for further details, see e.g. [4, 23]). The gauge covariant derivative is defined tion when writing di↵erential forms without indices—and I I I J as D↵ := d↵ + ! J ↵ , which in turns allows one to switch freely between these notations as and when is con- ^ I I I I J venient. define the Cartan torsion T = De = de + ! J e and the Cartan curvature RI = d!I +!I !K .We^ We now recall the vielbein formalism for NCT. Recent J J K ^ J investigations (see e.g. [23, 2.1]) have made it clear that now label these internal torsions and curvatures by their the gauge group of 4D NCT§ is the Bargmann group (just respective generators (cf. [4]): as the gauge group of first-order—i.e. ‘vielbein + spin connection’—GR is the Poincar´egroup), a (f)µ := Tµ (M)=2@[µm] 2![µ e]a, (7) 3 3+1 Barg (1, 3) := SO (3) nR n R U (1) , (1) ⇥ mass Tµ (H)=2@[µ⌧], (8) 3 a a ab a where Gal := (SO (3) ) is the homogeneous Galilean Tµ (P )=2@[µe] 2![µ e]b 2![µ ⌧], 0 nR group containing rotations and boosts, R3+1 represents (9) space and time translations, and the central extension a a ab Rµ (G)=2@[µ!] 2![µ !]b, (10) U(1)mass represents translations along a ‘mass dimen- ab ab sion’. The corresponding Bargmann algebra has the non- Rµ (J)=2@[µ!] . (11) zero commutation relations Since we are interested in Newtonian gravity, we will only a [Jab,Jcd]=4[a[cJd]b], (2) consider cases of NCT in which Tµ (H)=Tµ (P )=0 [Jab,Pc]= 2c[aPb], (3) (cf. [4, 7]). We further stress that the internal Cartan mass torsion f cannot be converted into a spacetime tor- [J ,G ]= 2 G , (4) ab c c[a b] sion because the extended vielbein is not invertible. [Ga,H]= Pa, (5) A Newton-Cartan (NC) spacetime is defined as µ µ [Ga,Pb]= abZ, (6) (M,h ,⌧µ), where M is a smooth 4-manifold, h := ab µ e ae b is a degenerate ‘spatial metric’ (which will be where Z is a central charge generating translations along used to raise indices), and the clock 1-form ⌧µ plays an internal ‘mass’ dimension, and H, Pa, Ga, and Jab the role of a ‘temporal metric’ (thus, the orthogonal- generate time translations, spatial translations, Galilean µ µ ity condition h ⌧ = 0 holds because e a is spacelike). boosts, and spatial rotations, respectively. Although hµ is non-invertible, one can still construct A Recall that in GR, a vielbein eµ is a geometrical the manifestly frame-dependent covariant tensor hµ := object which implements a change-of-basis so as to di- a b abe µe , which transforms under local Galilean boosts A B agonalise the metric field—we have gµ = eµ e ⌘AB. acting on frames. To discuss such frame-dependent quan- 1,d Since vielbeins are valued in the quotient algebra R ⇠= tities, it will be convenient to introduce the notion of a poinc/lor, they are associated with the gauge group of observer vector field, viz., a vector field nµ that is time- µ translations. By analogy with this familar construction, like and normalized (⌧µn = 1). 3

Following standard procedure, we can use the vielbein Teleparallel equivalent of GR (TPG)—We now review and the spin connection to equip NC spacetime with the modern approach to TPG [42]. Vielbein GR and 1,d a compatible spacetime connection (i.e., µh = TPG both use the Cartan data of an R -valued coframe r r A e I and a Lorentz spin connection ! I , albeit in dif- µ⌧ = 0), by means of the vielbein hypothesis µe + M MJ r A B e er ferent ways. In vielbein GR, one starts with the inertial !µBe = 0. Explicitly, has the Christo↵el symbols e r e degrees of freedom of special relativity (SR) and gener- n alizes to GR by ‘gauging’ the spin connection (i.e., treat- e µ = µ + ⌧µF , (12) ing it as a source of Cartan curvature), thus arriving at n the Levi-Civita spacetime connection which captures e r where n is an observer vector field, µ := n @(µ⌧) + both gravitational and inertial degrees of freedom. By 1 ⇢ a contrast, TPG restricts the spin connection to only rep- 2 h (@µh⇢ + @hµ⇢ @⇢hµ), and (F ) =(!a e ) is a 2-form. ^ resent the inertial e↵ects present in a particular coframe (12) shows us that unlike Lorentzian spacetime, NC and generalizes SR by ‘gauging’ the coframe to obtain spacetime does not induce a unique compatible torsion- torsion (which, as we mentioned above, is just the field free spacetime connnection. Instead, the space of pos- strength of the coframe). More precisely, TPG requires sible connections can be parametrized by fixing an ob- the spin connection to take the schematic form ! =⇤@⇤ n (where ⇤is a local Lorentz transformation), which is the server field n which determines , and then further most general condition under which the Cartan curvature specifying F in order to pick out µ.Thesetwo vanishes. In practice, it is often convenient to gauge-fix pieces of data have an important physical interpreta- I n to the case where !MJ= 0, thus restricting to the class tion: is an ‘inertial connection’ in thee sense that n is of frames (‘proper frames’) in which inertial e↵ects are n acceleration-free and vorticity-free with respect to , and absent. One can then use the vielbein hypothesis to de- µ µ fine the (unique) associated flat Weitzenb¨ock spacetime Fµ = ⌧µ↵ + µ ,where↵ := n n is the acceler- r • a b connection . We recall that the relationship between ation and µ := e [µe ]e aeb n is the vorticity of r r• r µ e of GR and of TPG is given by n with respect to . Finally, we will find it useful to r note that the spacetime boost connectione !µ := ! ae µ a • M M M can be written [23] e NP = NP + K NP , (16) M where K NP is the contorsion tensor. !µ = ✓µ + ⌧µ↵ + µ , (13) I Vacuum TPG is defined by the data M,eM , and a b µ solutions of the theory satisfy the equation of motion where ✓µ := e (µe )e aeb n is the expansion of n . r @ eS NM 4⇡ ej M =0, (17) In order to define classical NCT, the data N I I (M,e I ,! I ) must be supplementede with a mass µ µJ where e := det e I and the teleparallel gravitational density scalar field ⇢ and an observer field ⇠µ that M current, superpotential, and Lagrangian are respectively represents test particle trajectories, as well as the ‘New- given by: tonian condition’ dF = 0 and the vanishing rotational curvature condition R ab (J) = 0. We say that this 1 µ ej M = ee N S QM T P + e M , data is an NCT model if it satisfies the geodesic equation I 4⇡ I P QN I LG ⇠µ ⇠ = 0 and the source equation 1 rµ SMNP = KNPM gMPT QN + gMNT QP , 2 Q Q e⇣ ⌘ e Rµ =4⇡⇢⌧µ⌧. (14) = SMNPT , LG 16⇡ MNP On the other hand, the data of standard Newto- e • • nian gravitation (NG) is given by (M,hµ,⌧ , ,⇢,⇠µ,), where T M := M M is the torsion tensor, µ NP NP PN where is flat and compatible, and is a scalarr gravi- which encodes the antisymmetric part of the connection. tationalr potential; we say that this data is an NG model A straightforward calculation shows that (17) is equiv- µ if it satisfies the force equation ⇠ µ⇠ = and the alent to the vacuum Einstein equation RMN = 0. G is µ r r L Poisson equation µ =4⇡⇢. We can now state the equivalent to the Einstein-Hilbert Lagrangian up to a central foundationalr r result of NCT, viz., the Trautman boundary term—cf. [29, p. 92]. And under mild assump- Recovery Theorem [45]: given an NCT model, one can tions (in particular, covariant conservation of the source locally define an empirically equivalent NG model up to stress-energy tensor which would appear on the right the shift hand side of (17) in the non-vacuum case), it follows from application of the Mathisson-Papapetrou method for de- µ, µ + ⌧µ⌧ , + , (15) riving particle equations of motion [34] (cf. also [25]) that 7! r test particles satisfy the teleparallel force equation where µ = 0. (For a detailed discussion of this theoremr andr its implications, see [37, 44].) ⇠M • ⇠ = ⇠P T ⇠M . (18) rM N PNM 4

Clearly, there exists a striking analogy between the The name ‘twistless gauge’ stems from the observation flat NG connection in the Trautman recovery theorem that the vorticity of zµ vanishes with respect to , so our r r and the flat Weitzenbock connection • obtained through frame is twistless with respect to the NCT connection. r Since f = ⌧ d in this gauge, we see from (19)e that teleparallelizationGR. We now demonstrate that it is not ^ only possible to understand Trautman’s result in terms of µ µ µ an analogous teleparallelization , but that, remarkably, ⇠ µ⇠ =0 ⇠ µ⇠ = h (d)µ, (20) NC r () r this procedure commutes with teleparallelizationGR via null reduction. thus recoveringe the NG force law as the teleparallel equa- tion of motion. Newton-Cartan teleparallelization—In keeping with z the teleparallel philosophy, we separate out the gravita- Having gauge-fixed to the inertial connection µ,we tional and inertial degrees of freedom of NCT, by locating see from (13) that there is only one parameter left to the former in the ‘mass torsion’ and the latter in a suit- gauge-fix in !µ , viz. the expansion ✓µ .Wedosoby ably gauge-fixed spin connection. As above, tildes denote ab setting !µ = ✓µ = 0, which along with Rµ (J)=0 the NCT objects, whereas we omit any oversetting for the z associated teleparallel objects. (cf. the definition of an NCT model) implies that µ By analogy with the torsionlessness of GR, we require is flat. One can then easily deduce that the NCT source ˜ equation holds just in case the flat teleparallel connection that the mass torsion of NCT vanish, i.e. f = 0. From z (7), we thus see that the boost curvature is related to satifies the NG Poisson equation. a r the mass gauge field by dm˜ = !˜a e˜ . Note that the We have absorbed the Trautman recovery theorem into ˜ ˜ ˜ a ^ constraints f = Tµ (H)=Tµ (P ) = 0 allow one to teleparallelizationNC, but for the shift symmetry (15). In determine !˜I solely in terms of e˜I ,thusdeterminingthe fact, this symmetry is accounted for by the fact that the I I NCT spacetime connection . data (e¯ , ! J ) satisfying the twistless gauge and !µ =0 r We now introduce teleparallel NCT by identifying the is unique only up to U(1)mass gauge transformations that I I preserve these conditions. Cartan data (eM ,!MJ) thate yields (i) a flat spacetime connection (the analog of the Weitzenbock connection Null reduction commutes with teleparallelization— r of TPG), and (ii) whose Cartan torsion compensates for Having defined teleparallelizationNC, we show now that, the curvature of (the analog of the Levi-Civita connec- when intertwined with null reduction, this procedure r tion of GR). While one might be tempted to introduce commutes with teleparallelizationGR. In so doing, we non-trivial spatiale torsion, we note that for a flat connec- also give the first explicit teleparallel description of the a Bargmann-Eisenhart (BE) solution. tion, the Bianchi identities imply that Tµ (P ) = 0, so it is only the mass torsion f that is relevant here. The BE solution [7, 17] is a 5D vacuum solution of GR First, we perform a preliminary ‘inertial gauge-fixing’ equipped with a null vector field ⌅that is parallelized that is motivated by (12) and the discussion following it: by the Levi-Civita connection (and is thus Killing). It is I I well-known that 4D NC spacetime can be constructed as recall that (eM ,!MJ) determines the inertial connec- v the orbit space of the flow generated by ⌅, which is the tion µ precisely when it is subject to the constraint so-called ‘null reduction’. Since comprehensive discus- F = 0, upon which f = dm. Implementing this con- sions of this familiar technique have already been given straint and setting m˜ = m then yields the following re- in [17, 30], here we will only introduce the form that lationship between the NC and teleparallel equations of is best-suited to our computations, viz. the Julia-Nicolai motion: ansatz [30] for the BE solution, which directly incorpo- ⇠µ ⇠ =0 ⇠µ ⇠ = f ⇠µ. (19) rates null reduction by parameterizing the 5D relativis- µ µ µ M r () r tic frame fielde ˆ I in terms of 4D NC fields as follows: M µ µ Evidently, the tensor ⌧µF is the analogue of the contor- the only non-zero components ofe ˆ I aree ˆ a = e a, e v µ µ µ v µ v sion in this scenario, and it is determined by the telepar- eˆ a = e amµ,ˆe = v ,ˆe = v mµ, ande ˆ + = 1, allel mass torsion f. where the v spacetime-direction is aligned with ⌅. The I Clearly, the above gauge-fixing is not invariant under corresponding co-frame fielde ˆM has the non-zero com- a a + + local Galilean boosts. We can remedy this by trans- ponentse ˆµ = eµ ,ˆeµ = ⌧µ,ˆeµ = mµ, ande ˆv = 1. forming a vielbein to the ‘twistless gauge’ by means of (Note that as compared with [7, III], we set the com- a a a a a µa § e e¯ = e m ⌧ ,wherem = e mµ,thusre- pensating field S = 1.) We again stress that the power sulting7! in the new extended vielbein e¯I =(⌧ , e¯a, ⌧ ), of this ansatz stems from viewing the BE solution as an where is a boost-invariant scalar. The correspond- R-bundle over M (with fibers generated by ⌅) and mak- ing frame contains a boost-invariant observer field zµ := ing a judicious choice of section that lets us express the µ µ µ e¯0 = e0 h m. This choice is invariant under local 5D fields entirely in terms of the 4D NC fields contained I boosts acting on the unbarred quantities; however, there in eµ ; thus the null reduction can simply be read o↵the is still a residual boost-freedom that is parameterized by µ-components of the coframe field. Furthermore, at least the U(1)mass gauge transformations m m + df (where locally, we can use a di↵eomorphism to transform the f is an arbitrary smooth function), under7! which (zµ,) ansatz into the ‘twistless gauge’ where m = ⌧ and we µ is transformed into another boost-invariant pair (z0 ,0). will assume this gauge-fixing henceforth (cf. [30]); this is 5 of course consistent with the fact that the null reduction To relate this quantity to the teleparallel connection, of any BE solution automatically satisfies the Newtonian we recall that the Levi-Civita Ricci tensor can be written condition, as shown in [17]. We note that [8] showed that in terms of contorsion as (cf. [3, p. 3]) if the 5D GR fields are on-shell, their null reduced NC a fields will have Tµ (H)=Tµ (P ) = 0. Since we will R = KM KM + now construct the pure tetrad TPG version of the BE so- NQ rQ NM rM NQ R M R M lution, the spin connection vanishes and so this condition K NM K RQ K NQK RM . (29) amounts to d⌧ = dea = 0. Given the above ansatz, the two non-zero components Using (16), (24), (25), and mµ = ⌧µ, one finds for Rµ of the 5D Weitzenb¨ock connection read that

µ µ a µ • • = e @ e + z @ ⌧ , (21) Rµ = K = (⌧µ⌧h d). (30)  a ( ) ( ) r µ r v a  = e am@e @m + z m@⌧. (22) Combining (28) and (30) and using compatibility, we re- cover precisely the Poisson equation of NG, (Here, we have symmetrized indices in (21), since the tor- sion associated with this connection component vanishes µ • • h µ =4⇡⇢. (31) on-shell.) From this, one finds that all components of the r r torsion tensor vanish, save Finally, recent investigations have shown that TPG contains a gauge symmetry that can be expressed covari- v T  = 2@[m]. (23) antly as the ‘weak gauge transformations’ of [5], or from the Hamiltonian perspective as the -symmetry intro- The non-vanishing components of contorsion are (zµ is duced in [10]. When these TPG gauge transformations the boost-invariant observer field, defined above) preserve the above gauge-fixing, they give rise to the shift symmetry (15) of NG. µ µ µ K  = ⌧h @[m] ⌧h @[m], (24) Outlook.—The foregoing both o↵ers us deeper insight into the foundations of Newtonian gravitation, and opens Kv = ⌧ z@ m ⌧ z@ m 2@ m . (25)   [ ] [ ] [ ] up a rich array of possible future research directions. For (Some of these components are related to instance, it is well-known that NG faces problems of con- anholonomity—cf. [29, p. 39].) From these compo- sistency in infinite, homogeneous, isotropic universes (see e.g. [37, 4.4] and references therein). Roughly, one needs nents of contorsion, one can compute the components § of the superpotential SMNP, and thereby check that to pick a privileged centre point in any given solution— but then the magnitude of the gravitaitonal force is ar- G = 0 as expected, since we assumed at the outset that theL theory is on-shell. bitrary. The now-standard solution to this problem is the following: di↵erent solutions correspond to di↵erent To show that this teleparallelized theory just is NG, it gauge-fixings (which define NG) of a particular curved remains to derive the NG force equation for test particles, derivative operator of NCT, and consistency can be re- and the Poisson equation. For the former, recall that the stored by moving to the fully gauge-invariant picture of- only non-vanishing component of torsion is T v and so fered by NCT. the TPG force equation (18) yields We observe that an analogous consistency issue seems to arise in the case of a historic (but mistaken!) attempt M • v µ µ ⇠ M ⇠ = ⇠vT ⇠ = 2⇠v@[mµ]⇠ . (26) at defining TPG, i.e. the so-called ‘pure tetrad’ version of r µ TPG (see [31] for the original source, and [35] for contem- Using the gauge-fixing condition mµ = ⌧µ and metric porary discussion). Unlike modern TPG which is fully compatibility, we have gauge-invariant and consistent [42] (though see [38, 40] for some concerns regarding the Hamiltonian formulation µ • • ⇠ µ⇠ = , (27) of covariant TPG, and [9, 21] for replies), the pure tetrad r r version privileges a particular ‘proper frame’; our work which is the Newtonian force equation. shows that the inconsistency of NG is really a special To obtain the Poisson equation, we first note that we case of what happens when one does this. To see the a analogy explicitly, suppose we gauge-fix the spin connec- can write (14) in terms of the boost curvature Rµ (G), which in turn implies that R =4⇡⇢,whereRIJ = tion to vanish. Given this, we can determine a unique a M N K ab gµ; however, the tetrad field e is determined only up eˆ K eˆ I RMN J. If we then impose the Rµ (J)= µ a a b 0 condition, then it is easy to see (cf. [7, 3]) that the to a Lorentz transformation, e µ ⇤ e µ (here, Latin 7! b § indices are four-dimensional tangent space indices). This only non-zero component of RIJ is R ;usingRMN = I J is unproblematic for spinless matter, which couples to eˆM eˆN RIJ,wethushave the metric alone. However, in the pure tetrad theory, matter fields with spin couple to the tetrad directly— Rµ = ⌧µ⌧R =4⇡⇢⌧µ⌧. (28) so their behaviour will depend upon a choice of frame. 6

Thus, just as in the case of NG, we have an unpalat- ing conformal gravity, conformal TPG, twistless-torsional able gauge-dependence of the ‘force’ theory which may NCT, and the teleparallelization of twistless-torsional be resolved on moving to the associated ‘geometrized’ or NCT, in exactly the manner in which the diagram re- ‘covariantized’ theory. lating GR, TPG, NCT, and NG was constructed in this paper? We expect that doing so will be particularly use- Second, many applications of NCT to condensed mat- ful in applications in which it is helpful to make a judi- ter physics (e.g. [24]) use a generalization of NCT called cious choice of inertial frame, as is often the case when twistless-torsional NCT, which allows some degree of analyzing the behavior of fluids and lattices (cf. [24]). temporal torsion but still requires that the solutions re- main causal (in the sense that ⌧ d⌧ = 0, which en- sures the existence of hypersurfaces^ of absolute simul- ACKNOWLEDGEMENTS taneity). Furthermore, it was shown in [7] that twistless- torsional NCT can be constructed via the null reduction J.R. is supported by an AHRC scholarship at the Uni- of (on-shell) five-dimensional conformal gravity. Since versity of Oxford, and a predoctoral fellowship at the there exist well-known conformal generalizations of TPG University of Illinois at Chicago, under grant number (e.g. [6, 39]), our work paves the way for applying confor- 56314 from the John Templeton Foundation. N.T.’s mal TPG to condensed matter physics via a null reduc- work on this project was partially funded by NSF Award tion. In particular: can one construct a diagram, relat- 1734155.

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