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- efficient 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 = δab 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 λ = δ λ and eµ e B = δ 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, λ,... 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)µλ := Tµλ (M)=2@[µmλ] 2![µ eλ]a, (7) 3 3+1 − Barg (1, 3) := SO (3) nR n R U (1) , (1) ⇥ mass Tµλ (H)=2@[µ⌧λ], (8) 3 a a ab a where Gal := (SO (3) ) is the homogeneous Galilean Tµλ (P )=2@[µeλ] 2![µ eλ]b 2![µ ⌧λ], 0 nR − − group containing rotations and boosts, R3+1 represents (9) space and time translations, and the central extension a a ab Rµλ (G)=2@[µ!λ] 2![µ !λ]b, (10) U(1)mass represents translations along a ‘mass dimen- − ab ab sion’. The corresponding Bargmann algebra has the non- Rµλ (J)=2@[µ!λ] . (11) zero commutation relations Since we are interested in Newtonian gravity, we will only a [Jab,Jcd]=4δ[a[cJd]b], (2) consider cases of NCT in which Tµλ (H)=Tµλ (P )=0 [Jab,Pc]= 2δc[aPb], (3) (cf. [4, 7]). We further stress that the internal Cartan − mass torsion f cannot be converted into a spacetime tor- [J ,G ]= 2δ 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 − µλ µλ [Ga,Pb]= δabZ, (6) (M,h ,⌧µ), where M is a smooth 4-manifold, h := − ab µ λ δ 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 µλ µ ity condition h ⌧λ = 0 holds because e a is spacelike).