Disformal transformation in Newton–Cartan geometry
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Abstract
Newton–Cartan geometry has played a central role in recent discussions of the nonrelativistic holography and condensed matter systems. Although the conformal transformation in nonrelativistic holography can easily be rephrased in terms of Newton–Cartan geometry, we show that it requires a nontrivial procedure to arrive at the consistent form of anisotropic disformal transformation in this geometry. Furthermore, as an application of the newly obtained transformation, we use it to induce a geometric structure which may be seen as a particular nonrelativistic version of the Weyl integrable geometry.
Keywords
Conformal Transformation Universal Time Cartan Geometry Real Scalar Field Conformal Extension1 Introduction
Newton–Cartan geometry (NCG) was proposed by Élie Cartan as a geometrical description for Newtonian gravity in the spirit of general relativity [1, 2] (see also [3]). The recent renewed interest in NCG is well motivated in view of its usefulness in the investigations of condensed matter systems and nonrelativistic holography. Concretely speaking, it has been found that, together with the nonrelativistic diffeomorphism invariance, NCG provides a natural geometrical background for the effective field theory description of fractional quantum Hall effect [4].
Moreover, the progress in gauge/gravity duality attaches increasing importance to nonRiemannian geometries while, in particular, NCG acts an important role in various realizations of the nonAdS holography; see e.g. [5, 6, 7, 8, 9].
Similar to the extension of Riemannian geometry to Weyl geometry, the conformal extension of NCG has also been investigated from different perspectives. Following the experience of “deriving” Einstein gravity (more properly, Einstein–Cartan theory) and Riemann–Cartan geometry through gauging the relativistic Poincaré algebra, NCG is obtained by gauging the Bargmann algebra, which is the centrally extended Galilean algebra [10]. Then the conformal generalization of NCG is also got by performing a similar gauging procedure to the Schrödinger algebra, the conformal version of Bargmann algebra [8, 11]. On the other hand, inspired by the celebrated Poincaré gauge theory [12, 13, 14] (see [15] for a detailed exposition) in which Riemann–Cartan geometry emerges naturally by localizing the global Poincaré symmetry of a field theory in Minkowski spacetime, it has been shown that NCG can be obtained by localizing the global Galilean symmetry of a field theory in 3d Galilean space with universal time [16]. Such a method also provides a systematic way for reformulating an originally Galileaninvariant theory in Euclidean space with universal time into a diffeomorphisminvariant theory in curved space [16, 17, 18]. Following this train of thought, the conformal extension of NCG is also obtained by localizing the global Galilean and scale symmetries [19].
The above two methods are not merely straightforward repetitions of the corresponding process in the relativistic case. There the space and time are on an equal footing and thus can be treated uniformly which makes the whole procedure relatively clear and concise. However, for nonrelativistic cases, the space is relative while the time is not [20, 21]. To preserve the absoluteness of universal time and the relative character of space, special attention is needed throughout the whole procedure; see [10, 16, 17, 18] for details. Such different concepts of space and time are also reflected in the consistent form of the conformal transformation. In relativistic cases, conformal transformation must be isotropic due to the equal footing of space and time. Nevertheless, in nonrelativistic case, the absolute nature of the universal time makes the concept of anisotropic conformal transformation emerge naturally; see [8, 11, 19] for further details.
The disformal transformation has attracted a lot of attention in recent years and found applications in e.g. varying speed of light models [25], inflation [26], dark energy models [27] and dark matter models [28]. In particular, it acts an important role in Horndeski theories and has led to many important results in modified gravity and cosmology [23, 24, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. It is interesting to note that there are very few discussions as regards disformal transformation in the nonrelativistic realm. Even the related work on modified Newtonian dynamics paradigm has focused on the relativistic extensions [41]. Except for the obvious applications in gravity theories, a possible nonrelativistic version of the disformal transformation may have implications for the nonrelativistic holography given the power and utility of conformal transformations in this area.

Is there a consistent nonrelativistic version of the disformal transformation?

In view of the relation between conformal/disformal transformation and Weyl integrable geometry [42], one may wonder that can the NCG be disformally generalized?
Simply speaking, the disformal extension of NCG cannot easily be achieved by the aforementioned two methods. This fact makes it an interesting problem from a mathematical point of view. What is more, a sound knowledge about the nonrelativistic disformal transformation and disformal extension of NCG is also helpful for constructing disformally invariant field theories as well as gravity theories, which have the potential to enrich the investigations broaden the horizons of condensed matter theory and holography. All these considerations constitute the motivation for the present work. First of all, we will show that a consistent form of disformal transformation in NCG can indeed be given (see (30)). After that, we will use the newly obtained disformal transformation to induce a new geometry, not by gauging the disformal extension of Bargmann algebra or localizing the global symmetry of a disformally invariant field theory in NCG, but by using the method which has been used in [42] to get the disformal extension of Riemannian geometry.
This paper is organized as follows. In Sect. 2, a brief review of the general torsional NCG recently expatiated in [11] is given. We also demonstrate that the conformal extension of NCG can be induced through the anisotropic Weyl rescaling of original metrics. This serves as a complementary perspective to the standard gauging or localization procedure. In Sect. 3, we propose a method to find the most general anisotropic disformal transformation in NCG. Then, as an application of the newly obtained disformal transformation, we use it to induce the disformal extension of NCG in Sect. 4. The results consist of the basic disformal connection and semimetricity conditions. Our conclusion will be given in the last section.
2 Newton–Cartan geometry and its conformal extension
2.1 Newton–Cartan geometry
A first taste of NCG can be gained through a geometric reformulation of Newtonian gravity; see [43] for a clear exposition. Recently it has been shown that NCG can be obtained by gauging the centrally extended Galilean algebra, i.e., the Bargmann algebra [8, 10, 11]. A wellknown fact is that general relativity can be obtained by gauging the Poincaré algebra. This method will naturally lead to the appearance of torsion and result in Riemann–Cartan geometry. Only after one imposes zero constraints on the curvature tensors of gauge fields related to generators of translational symmetries, then the torsionfree Riemannian geometry can be obtained. In nonrelativistic cases, the situation is similar: gauging the Bargmann algebra will also lead to the appearance of torsion; however, after imposing several curvature constraints, the final NCG will have no torsion [10]. On the contrary, if no zero constraints were imposed, the geometry one obtained will be the torsional NCG [11].
Torsional NCG acts an important role in nonAdS holography. For example, the boundary geometry in Lifshitz holography is described by torsional NCG [7, 9]. For related studies on supergravity in NCG with or without torsion, see [44, 45, 46, 47, 48]. All in all, whether there is torsion or not can lead to very different results. While the dynamical NCG without torsion is related to projectable Hořava–Lifshitz gravity [49], the incorporation of twistless torsion leads to nonprojectable Hořava–Lifshitz gravity [11]. Since the realization of disformal transformation in NCG and its potential applications are of our most concern, we would like to maintain as general as possible. In other words, no gauge fixing on the NCG will be imposed and a general torsion (with no twistless condition) is assumed. For our purpose, it is convenient to make use of the formalism of dynamical NCG developed in [11]. To begin with, let us list some relevant results which will be the starting point of our discussion.
2.2 Conformal extension of Newton–Cartan geometry
In principle, via conformal rescaling (which reduces to Weyl rescaling in the isotropic case), the conformal NCG could be induced from the original geometry. Our main task in this subsection is to demonstrate this point in detail.
We remark that the conformal extension of NCG, whose metric compatibility conditions and connection are given by (15), (16) and (17), respectively, can be seen as a nonrelativistic version of the Weyl geometry (hence the superscripts “W”s in the above equations). Our results are also consistent with those obtained through the gauging procedure [8, 11]. Furthermore, when \(b_\mu \) (which is just the dilatation parameter in [8, 11]) can be expressed as the derivative of a scalar field \(\partial _\mu b\), the corresponding geometry defined by (15)–(17) is just the Weyl integrable extension of NCG, whose relativistic cousin is the wellknown Weyl integrable geometry.
To sum up, the defining relations (15)–(17) can be obtained through the rewriting and rearrangement of the original variables and expressions. It is just in this sense that we say the conformal extension of NCG can be induced from NCG by conformally rescaling the metrics. The potential advantage of this perspective is that one can use this approach to induce a new geometry even when the symmetries and algebraic structures of the underlying geometry are not well understood yet. This method has recently been utilized to induce a new geometry from Riemannian geometry through the disformal transformation [42].
3 Anisotropic disformal transformation in Newton–Cartan geometry
It should be noticed that the quantity \(\alpha \hat{v}^\mu \hat{v}^\nu +\beta \hat{h}^{\mu \nu }\), with special nonzero parameters, is sometimes referred to as the extended metric and has arisen frequently in the literature [3, 9, 11, 51, 55]. Such an extended metric is a nondegenerate symmetric rank 2 tensor with Lorentzian signature. However, this metric should not be treated as a fundamental dynamical variable in NCG. Indeed, if a single nondegenerate metric could be defined in NCG, there would be no need for introducing two degenerate metrics. It is precisely the lack of such a nondegenerate metric that led Cartan to formulate NCG in this way.

The first two expressions in (30) suffice to give the definition of the corresponding disformal transformation. This is because their inverses can be derived from them and thus are not independent.

The previous result in (25) simply corresponds to the special case when \(Y=\frac{1}{2} F\). Obviously in this case \(\phi _\mu \) has only a temporal component.

With respect to the disformal transformation (18) in relativistic gravity theories, it has been shown that \(A(\phi ,X)\) must be positive definite, and \(B(\phi ,X)\) should satisfy some constraints to ensure a healthy definition [22, 23, 24]. The similar situation is also expected to happen in nonrelativistic cases. Nevertheless, this is not of the present concern and is left for future study.
4 Disformal extension of Newton–Carton geometry
As an application of the newly obtained disformal transformation (30), we would like to use it to induce a new geometry which is a disformal extension of NCG. Noticing that the relativistic disformal transformation in the form \(\bar{g}_{\alpha \beta }=A(\phi )g_{\alpha \beta }+B(\phi )\phi _{\alpha }\phi _{\beta }\) is of particular interest in recent literature [23, 24, 34, 35, 36, 37, 38, 39, 40], we will restrict our attention to the special case where the disformal parameters A, B, and C in (30) are only functions of the scalar field \(\phi \) throughout this section.
 Disformal connection:Here \(\partial _\mu \partial _\mu \big (\ln \frac{A^z}{1+2B Y} \big )\) can be interpreted as the Weyl covariant derivative along the time direction, and \(\partial _\mu \partial _\mu \ln A^2\) is the Weyl covariant derivative on the spatial slices. For the special case with \(B=D=0\), (42) reduces to (17), which is the result obtained in the conformal extension of NCG.$$\begin{aligned} ^D\Gamma ^\lambda _{\mu \nu }= & {}  \hat{v}^\lambda \bigg [ \partial _\mu \partial _\mu \Big ( \ln \frac{A^z}{1+2B Y} \Big ) \bigg ] \tau _\nu \nonumber \\&+ \, \frac{1}{2} \Big ( h^{\lambda \sigma }+\frac{D}{12D Z} \Phi ^\lambda \Phi ^\sigma \Big ) \Big [(\partial _\mu \nonumber \\& \, \partial _\mu \ln A^2)(\hat{h}_{\sigma \nu }D\Phi _\sigma \Phi _\nu )\nonumber \\&+ \, (\partial _\nu \partial _\nu \ln A^2)(\hat{h}_{\mu \sigma }D\Phi _\mu \Phi _\sigma )\nonumber \\& \, (\partial _\sigma \partial _\sigma \ln A^2)( \hat{h}_{\mu \nu }D\Phi _\mu \Phi _\nu )\Big ] . \end{aligned}$$(42)
 Semimetricity conditions:$$\begin{aligned}&^D\nabla _\mu \tau _\nu = \partial _\mu \bigg (\ln \frac{A^z}{1+2B Y}\bigg )\cdot \tau _\nu , \end{aligned}$$(43)These two semimetricity conditions together with the disformal connection (42) define the disformal extension of NCG. Note again that the conformal extension of NCG (16) is naturally included here as a special case. Furthermore, the geometry defined by (42)–(44) is obviously a nonrelativistic and a disformally extended version of the conventional Weyl integrable geometry.$$\begin{aligned}&^D\nabla _\mu \Big ( h^{\alpha \beta } + \frac{D}{12D Z} \Phi ^\alpha \Phi ^\beta \Big )\nonumber \\&\quad =\partial _\mu \ln A^2\cdot \Big ( h^{\alpha \beta } + \frac{D}{12D Z} \Phi ^\alpha \Phi ^\beta \Big ). \end{aligned}$$(44)
5 Conclusion and discussion
In this work, we have found the general anisotropic disformal transformation (30) in NCG. It is shown that a naive assumption of the form of the disformal transformation can only lead to a very special form which is consistent; see (25). To obtain the most general form, one needs to project the vector \(\phi _\mu \) in the relativistic disformal transformation (18) into temporal and spatial parts, and then impose the constraints from the defining relations in NCG.
As an application, this newly obtained disformal transformation has been used to induce a new geometry whose disformal connection and semimetricity conditions are given in (42) and (43, 44), respectively. The key step to do this is to rewrite the original geometrical variables with new variables obtained through disformal transformation. The corresponding subtle point is an infinite iteration in rewriting \(h^{\mu \nu }\) with \( h^{\mu \nu }\) and \(\Phi ^\mu \) which is due to the hidden \(h^{\mu \nu }\) in \(\Phi ^\mu \). This problem has been circumvented by using another equivalent (but more convenient) form of the disformal transformation; see (35).
For further investigation, it would be interesting to see whether the nonrelativistic and anisotropic disformal transformation (see (30)) can be reproduced through holography. Concretely speaking, if the holography relates a bulk relativistic geometry to a boundary nonrelativistic one and the disformal transformation for the bulk relativistic gravity is known, then one could in principle use this holography to obtain the corresponding anisotropic disformal transformation induced on the boundary. Second, it may also be interesting to figure out the disformal extension of Poincaré and Bargmann algebra exactly, then to obtain the disformal extension of Riemann–Cartan geometry, as well as NCG, via a gauging procedure. Third, as hinted at in the introduction, our results may be related to modified Newtonian dynamics paradigm [41, 56] and other modified gravity theories. Finally, in view of the role of conformal transformation and Schrödinger algebra in nonrelativistic holography as well as the recent discussions as regards disformal transformation in cosmology, one may speculate the possible utility of disformal transformation in Lifshitz holography [57] and holographic cosmology [58, 59].
Footnotes
 1.
For a detailed explanation of these quantities (as well as \(\tilde{\Phi }\), which will appear soon) and their relevance in recent literature, see [11] and the references therein. See also [10, 50] for torsionfree cases while other aspects of NCG have been discussed in [50]. For present purposes, it suffices to content ourselves with the minimal introduction of NCG.
Notes
Acknowledgments
We sincerely thank the anonymous referees for valuable suggestions.
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