# Embedding theory as new geometrical mimetic gravity

## Abstract

It is well known that the recently proposed model of mimetic gravity can be presented as general relativity with an additional mimetic matter. We discuss a possibility to analogously reformulate the embedding theory, which is the geometrical description of gravity proposed by Regge and Teitelboim, treating it also as general relativity with some additional matter. We propose a form of action which allows one to describe this matter in terms of conserved currents. This action turns out to be a generalization of the perfect fluid action, which can be useful in the analysis of the properties of the additional matter. On the other side, the action contains a trace of the root of the matrix product, which is similar to the constructions appearing in bimetric theories of gravity. The action is completely equivalent to the original embedding theory, so it is not just some artificial model, but it has a clear geometric meaning. We discuss the possible equivalent forms of the theory and ways of studying the equations of motion that appear.

## 1 Introduction

*mimetic gravity*was proposed 5 years ago in Ref. [1] and became quite popular in recent years. In this framework the conformal degree of freedom of gravity is isolated by introducing a parametrization of the physical metric in the following form:

*mimetic matter*, one can say that “‘dark matter’ without dark matter, which is imitated by extra scalar degree of freedom of the gravitational field” [1] arises in the theory.

*pressure-free perfect fluid*with

*potential*motion. It allows one to write an equivalent formulation of mimetic gravity, where the physical metric is considered to be independent, and an additional action term corresponding to such a perfect fluid is present. For the first time this was done in Ref. [2], where the full action is chosen to be

*n*can be treated as the number density of mimetic matter particles and \(\lambda \) defines their velocity \(u_\mu =\partial _\mu \lambda \). There are other ways of choosing the additional term in the action for a potentially moving pressure-free perfect fluid; see [3].

There are also many other ways to write an action for mimetic gravity; see, e.g., [4, 5].

The framework of mimetic gravity can be modified to explain certain phenomena of modern cosmology. In particular, the introduction of a potential for the scalar field \(\lambda \) [6] leads to the appearance of a pressure of the mimetic matter (note that the structure of the action used appears to be a particular case of the general expression which was already considered in the earlier work [7]), whereas the addition of a higher-order derivative term transforms the mimetic matter into an imperfect fluid [6, 8, 9]. It is also possible to write an action of the mimetic matter in a form with which it turns out to be pressure-free perfect fluid moving arbitrarily (i.e. not potentially anymore) [3]. In the latter case the description of the mimetic matter contains two more fields in addition to *n* and \(\lambda \). Moreover, one can return to the original formulation of mimetic gravity with the auxiliary metric \(\tilde{g}_{\mu \nu }\) and three scalar fields that are present in the expression for physical metric \(g_{\mu \nu }\) analogous to (1); see for details [3]. For the current status of the mimetic gravity framework see the review [10] and the references therein.

The transformations (1) on which the mimetic gravity is based are a special case of the more general “disformations” [11], which are invertible in the general case, i.e. the auxiliary metric \({\tilde{g}}_{\mu \nu }\) can be expressed through the physical metric \(g_{\mu \nu }\) and the scalar field \(\lambda \). It leads to the fact that in the general case of such a change of variables the theory turns out to be equivalent to GR [12]. On the contrary, the special transformations (1) are non-invertible because the conformal mode of \({\tilde{g}}_{\mu \nu }\) does not contribute to \(g_{\mu \nu }\), i.e. the conformal invariance appears [4]. Hence the mimetic gravity is an example of a theory modification appearing as a result of the change of variables that contains differentiation, while the number of variables remains unchanged as 10 components of \(g_{\mu \nu }\) is replaced by 9 components \({\tilde{g}}_{\mu \nu }\) (without the conformal mode) and the scalar \(\lambda \).

This modification is caused by the change of the class of variations of independent field variables, which alters the set of possible solutions of the theory according to the property of the variational principle. In the case of the change of variables that do not contain a differentiation, the change of the class of variations does not occur (of course, if the change of variables does not affect the number of degrees of freedom), so the theory remains unchanged. An example of such a case is a transition from the metric formulation of GR to the tetrad one. Also, if the change of variables contains only spatial derivatives, then the theory usually is not affected due to the common assumption that at spatial infinity the fields decrease sufficiently fast. Therefore the presence of time derivatives in the change of variables leads to a significant modification of the theory. Since the class of variations becomes smaller after such a change (the variation of an independent variable is always assumed to vanish at the initial and final moments of time, so this requirement poses an additional restriction on the original variable), the set of solutions of the modified theory turns out to be larger, including all the solutions of the original theory. It is precisely so in the case of mimetic gravity: the solutions of (3) at \(n=0\) (which corresponds to the absence of mimetic matter) are the ones of GR. A simple example of a change of variables with differentiation in 0-dimensional (i.e. mechanical) theory can be found in [13].

*embedding theory*. In contrast with mimetic gravity (1) the underlying change of variables in the embedding theory approach has a deep geometric meaning. In this string-inspired approach the assumption is made that our spacetime is not only an abstract pseudo-Riemannian space, but rather a surface in a flat

*N*-dimendional ambient space (bulk). Consequently, the induced metric appears on the surface

*extra*solutions. After the original paper [14] the ideas of embedding theory were discussed in detail in [16], and later were used in numerous papers devoted to the various aspects of gravity, including quantization; see, e.g., [17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. A detailed bibliography of embedding theory and closely related topics can be found in [27].

*embedding matter*:

Several questions can be raised: how should one write an action of the embedding theory in the form of GR with additional embedding matter, what is this matter like, and what are its laws of motion from the physical point of view? The most straightforward way to answer the first question was proposed in [3], but the choice of independent variables made there does not allow one to say anything related to the second question. In the present work we will consider some alternative ways of choosing the action of embedding theory, as well as of choices of independent variables for the description of embedding matter which is more convenient in a discussion of its physical meaning.

## 2 Forms of action for embedding matter

^{1}However, the physical meaning of the variables \(\tau ^{\mu \nu }\) and \(y^a\) which describe the embedding matter in such an approach remains unclear.

**tr**means subsequently taking the trace. The independent variables describing the embedding matter in this approach (which we will call

*embedding gravity*) are the quantities \(j^\mu _a\) and \(y^a\).

*I*is an identity matrix. Then for the variation we have

*A*(under the assumption that the root branch is taken so that \(\sqrt{I}=I\)), one can easily notice that the symmetry of matrix \((g^{-1}A)\) together with the symmetry of

*g*leads to the symmetry of \((g^{-1}\sqrt{I+A})\) as all the terms in the series in this case are symmetric. As a result we get \(\beta ^{\nu \alpha }=\beta ^{\alpha \nu }\), so the inverse of it (which we will denote \({\hat{\beta }}_{\mu \nu }\): \({\hat{\beta }}_{\mu \nu }\beta ^{\nu \alpha }=\delta _\nu ^\alpha \)) will also be symmetric: \({\hat{\beta }}_{\mu \nu }={\hat{\beta }}_{\nu \mu }\). Using this fact together with (17), we obtain the equation of motion as the result of variation of (13) w.r.t. \(j^\mu _a\):

*a*can be treated as a conserving (according to (14)) current density of a certain type of matter, whereas the \(y^a\) turn out to be Lagrange multipliers providing this conservation. Such a form of the action with the Lagrange multiplier, providing current density conservation

An interesting observation is that the structure appearing in the action (13) of the embedding matter, namely the square root of the matrix product, is well known by the so-called bimetric gravity theories [29]. In these theories the role of the matrices is played by two independent metrics, whereas in the action (13) one matrix is a metric \(g_{\mu \nu }\) and the other is a symmetric quadratic expression \(j^\nu _a j^{\alpha a}\). Among the different topics of study in bimetric theories the question of using non-standard root branches is considered along with other mathematical aspects connected with the presence of such a singular expression in the theory; see, e.g. [30, 31]. Such subtleties in the approach of embedding gravity require additional study.

## 3 Conclusion and possible development

Starting from the geometric description of the formulation of gravity proposed by Regge and Teitelboim, the *embedding theory*, one can rewrite the theory in the form of *embedding gravity* (11), (13) which is GR with some additional matter. Such a transition is analogous to *mimetic gravity* (2), (1), when a theory, which was initially formulated as a result of certain change of variables in GR, is rewritten as GR with additional matter resembling a perfect fluid with no vorticity (5), (6). Both approaches can be used in attempts to solve the dark matter and dark energy problems: for the mimetic gravity approach, see [10]; for the embedding theory approach this question was studied in the FRW approximation in Refs. [32, 33, 34].

The approach of the embedding theory has a clear geometrical meaning: spacetime is treated as a surface in the flat bulk, whereas the mimetic gravity is based on the change of variables (1), which is constructed artificially to separate the conformal mode of the metric. In the original formulation of mimetic gravity the appearing fictional matter has very simple properties (it is a perfect fluid with potential motion), so the model must be modified (e.g. by introduction of additional parameters; see the Introduction) to explain dark matter. On the contrary, the matter appearing in the embedding gravity approach is highly nontrivial by itself, though it has some properties in common with a perfect fluid. Therefore the attempt to explain dark matter (and possibly dark energy) through embedding gravity with the above action seems promising. An especially interesting problem is the construction of the above-mentioned (see before (23)) “microscopic” description of the embedding matter, which could automatically provide satisfaction of continuity equation (14) for a matter current.

*a*(e.g. \(j^\mu _a=j^\mu \delta ^0_a\)), then the embedding matter precisely becomes a perfect fluid; see after (22). Therefore one can analyze the equations of embedding gravity in the framework of a “non-relativistic” approximation from the point of view of the ambient space (for each value of the \(\mu \) ambient space vector \(j^\mu _a\) its 0th component prevails):

This nontriviality is largely related to the non-linearizability of the Regge–Teitelboim equation (8) (their properties were discussed, e.g., in [13]) in the original formulation of the embedding theory when the most natural background embedding function \(y^a(x)\) is chosen which corresponds to a 4D plane. Also note that because of this non-linearizability it is difficult to use the results obtained for the dynamics of the embedding matter in the framework of the FRW symmetry [32, 33, 34] as a basis on the perturbation theory aimed to transcend this symmetry.

*T*is the brane tension) with any kind of \(S^{\text {add}}\) discussed in Sect. 2, one obtains the description of a 3-brane; and lowering the dimension from 4 to 2 leads to the Nambu–Goto bosonic string. To prove this, one needs to write the equation of motion of the brane described by the embedding function \(y^a(x)\) with the action \(S^{\text {b}}[g_{\mu \nu }]\), where the metric is given by (7), in the form analogous to (10):

## Footnotes

## Notes

### Acknowledgements

The authors are grateful to A. Golovnev and A. Starodubtsev for useful discussions. The work of one of the authors (A. A. Sheykin) was supported by RFBR Grant N 18-31-00169.

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