Physics Research International

Volume 2015, Article ID 152846, 19 pages

http://dx.doi.org/10.1155/2015/152846

## Modified Theories of Gravitation behind the Spacetime Deformation

Ambartsumian Byurakan Astrophysical Observatory, Aragatsotn, 378433 Byurakan, Armenia

Received 6 June 2014; Revised 1 September 2014; Accepted 11 September 2014

Academic Editor: Bennie F. L. Ward

Copyright © 2015 Gagik Ter-Kazarian. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In the framework of proposed theory of spacetime deformation/distortion, we have a way to deform the spacetime through a nontrivial choice of the distortion-complex, displaying different connections, which may reveal different post-Riemannian spacetime structures as corollary. We extend this theory to address, in particular, the gauge model of the most general metric-affine gravity carrying both nontrivial torsion and nonmetricity. This model is constructed in the framework of the first order Lagrangian expressed in terms of the gauge potentials and their first derivatives. The equations of the standard theory, which have no propagating modes for torsion, can be equivalently replaced in modified framework by the modified equations, which in the limit of reducing the affine group leads to the modified Einstein-Cartan theory with dynamical torsion and beyond. In testing the modified framework for various particular cases, we use the Lagrange multipliers for extinguishing nonmetricity and torsion.

#### 1. Introduction

We will not attempt a history of gauge theory of gravitation in general, but only of those that seem most relevant to the particular theory of this paper. From its historical development, the efforts in gauge treatment of gravity mainly focus on the quantum gravity and microphysics, with the recent interest, for example, in the theory of the quantum superstring or, in the very early universe, in the inflationary model. The papers on the gauge treatment of gravity provide a unified picture of gravity modified models based on several Lie groups [1–61]. However, currently no single theory has been uniquely accepted as the convincing gauge theory of gravitation which could lead to a consistent quantum theory of gravity. They have evoked the possibility that the treatment of spacetime might involve non-Riemannian features on the scale of the Planck length. This necessitates the study of dynamical theories involving post-Riemannian geometries. On the other hand, a general way to deform the spacetime metric with constant curvature has been explicitly posed by [62–66]. The problem was initially solved only for 3D spaces, but consequently it was solved also for spacetimes of any dimension. It was proved that any semi-Riemannian metric can be obtained as a deformation of constant curvature matric, with this deformation being parameterized by a 2-form. A novel definition of spacetime metric deformations, parameterized in terms of scalar field matrices, is proposed by [67]. However, without care of the historical justice and authenticity, it also should be emphasized that the continuous deformations within the space of connections were studied by [17, 18]. This concept of a deformation seems to be a special case of a prolongation. A main idea is as follows: since any tensor-valued one-form transforms homogenously with respect to linear gauge transformations, its subtraction from a connection can be regarded as a continuous deformation within the space of connections. A more general approach is suggested by [68], whereas the authors present the extension to 4D of an euclidean 2D model that exhibits spontaneous generation of a metric. In this model gravitons emerge as Goldstone bosons of a global symmetry broken down to . They formulated microscopic theory without having to appeal to any particular spacetime metric and only assume the preexistence of a manifold endowed with an affine connection. Moreover, here it seems that not even a flat metric needs to be assumed. In this sense the microscopic theory is quasitopological. The vierbein appears as a condensate of the fundamental fermions. In spite of having nonstandard characteristics, the microscopic theory appears to be renormalizable. Recently, some clarifications are also collected in the book [69], where the authors have tried to collect the established results and to focus new investigations on the real loopholes of the theory. Only time will tell whether any of these intriguing theories is correct and which of the hypothesized hidden symmetries is actually realized in nature.

It is well known that the notions of space and connections should be separated; see, for example, [13, 70–72]. The curvature and torsion are in fact properties of a connection, and many different connections are allowed to exist in the same spacetime. Therefore, when considering several connections with different curvature and torsion, one takes spacetime simply as a manifold and connections as additional structures. Allowing minimal departure from semi-Riemannian geometry would consist in admitting torsion. The concept of a linear connection as an independent and primary structure of spacetime is the fundamental proposal put forward by Cartan’s geometrical analysis [43–46], whereas he gave a beautiful geometrical interpretation of torsion representing a translational misfit. The Einstein-Cartan (EC) theory also called Einstein-Cartan-Sciama-Kibble theory is the minimal extension of the general relativity, which considers curvature and torsion as representing independent degrees of freedom. In the standard EC theory, the equation defining torsion is of algebraic type, and not a differential equation, and no propagation of torsion is allowed. Even at the textbook level it was common knowledge known from the weak interaction that the causality reasons do not respect a contact interaction. Therefore, many modifications of the EC theory have been proposed in recent years; see, for example, [37, 73–79], but all these approaches are subject to many uncertainties. In particular, in [80] the authors analyze the functional renormalization group flow of quantum gravity on the EC theory space. The latter consists of all action functionals depending on the spin connection and the vielbein field (coframe) which are invariant under both spacetime diffeomorphisms and local frame rotations. They find evidence for the existence of at least one non-Gaussian renormalization group fixed point which seems suitable for the asymptotic safety construction in a setting where the spin connection and the vielbein are the fundamental field variables. The difference between Einstein’s general relativity and its Cartan extension is analyzed within the scenario of asymptotic safety of quantum gravity by [81], whereas it was found that the four-fermion interaction distinguishes the Einstein-Cartan theory from its Riemannian limit. However, modifications of EC theory are intricate. In a recent paper [82], we construct the* two-step spacetime deformation* (TSSD) theory which generalizes and, in particular cases, fully recovers the results of the conventional theory of spacetime deformation [62–66]. Conceptually and techniquewise the TSSD theory is versatile and powerful and manifests its practical and technical virtue in the fact that through a nontrivial choice of explicit form of a* world-deformation tensor*, which we have at our disposal, in general, we have a way to deform the spacetime displaying different connections, which may reveal different post-Riemannian spacetime structures as corollary: (1) the Weitzenböck spacetime structure—() underlying a teleparallelism theory of gravity, see, for example, [23, 24, 38, 83–93]; (2) the RC manifold—() underlying EC theory, for a comprehensive references, see, for example, [34–37, 51, 52, 94]; (3) or even the most general linear connection of metric-affine gravity (MAG) taking values in the Lie algebra of the 4D-affine group, ; see, for example, [15–18, 42–50, 55–57]. This represents the semidirect product of the group of 4D-translations and general linear 4D-transformations. Continuing along this line, in the present paper, we address the essential features of the gauge model of the most general MAG theory in context of TSSD-construction of post-Riemannian geometry, whereas to complete the TSSD theory, we build up the* distortion-complex* (DC) and show how it restores the* world-deformation tensor*, which still has been put in [82] by hand. A formulation of the major physical aspects of the modified gauge MAG theory will be given in the framework of the first order Lagrangian expressed in terms of the gauge potentials and their first derivatives. All the fundamental gravitational structures in fact, the metric as much as the coframes and connections, acquire a DC induced theoretical interpretation. There is another line of reasoning which supports the side of this model. We address the key problem of a* dynamical torsion* and show that the equations of the standard MAG theory can be equivalently replaced by the set of* modified MAG equations* in which the torsion, in general, is turned out to be* dynamical*. We define the physical conditions for the spacetime deformations when the spin-spin interaction becomes short-range propagating. As an application we have to test the general TSSD-MAG framework in some limit, namely, we have to put on Lagrange multipliers to recover the TSSD versions of different (sub)cases of Poincaré gauge theory (PG), Einstein-Cartan theory, teleparallel gravity (), and general relativity (GR). This allows amplifying and substantiating the assertions made in [82]. Moreover, imposing different physical constraints upon the spacetime deformations, in this modified framework we may reproduce the term in the well known Lagrangian of pseudoscalar-photon interaction theory, or terms in the Lagrangians of pseudoscalar theories [95–100]. This paper is organized as follows. To start with, in Section 2 we complete the spacetime deformation theory [82] by introducing DC complex and showing how it restores the* world-deformation tensor*. We also briefly revisit the theory of TSSD to make the rest of the paper understandable. An outline of the key points of TSSD theory of relevance to gauge MAG theory is stated in Section 3. The concluding remarks are presented in Section 4. The appendices discuss some relevant topics on the TSSD theory and algebraic operations in use.

#### 2. Spacetime Deformations and Beyond: Model Building

Let us consider a smooth deformation map , written in terms of the* world-deformation* tensor , general, , and flat, , smooth differential 4D-manifolds. The tensor, , can be written in the form , where the distortion-complex (DC) includes the invertible distortion matrix and the tensor and ). Here we use Greek alphabet () to denote the holonomic world indices related to and the second half of Latin alphabet () to stand for the world indices related to . The principle foundation of a* world-deformation* tensor comprises the following two steps [101]: (1) the basis vectors at given point () undergo* distortion* transformations by means of the matrix ; and (2) the diffeomorphism is constructed by seeking a new holonomic coordinates as the solutions of the first-order partial differential equations. Namely,
where the conditions of integrability, , and nondegeneracy, , necessarily hold [102, 103]. For reasons that will become clear in the sequel, we write the norm of infinitesimal displacement on in terms of the spacetime structures of :
The holonomic metric on the space can be recast in the form , with the components in dual holonomic basis . In order to relate local Lorentz symmetry to more general deformed spacetime, there is, however, a need to introduce the soldering tools, which are the linear frames and forms in tangent fiber bundles to the external general smooth differential manifold, whose components are the so-called tetrad (vierbein) fields. The has at each point a tangent space, , spanned by the anholonomic orthonormal frame field, , as a shorthand for the collection of the 4-tuplet , where . We use the first half of Latin alphabet () to denote the anholonomic indices related to the tangent space. The frame field, , then defines a dual vector, , of differential forms, , as a shorthand for the collection of the , whose values at every point form the dual basis, such that , where denotes the interior product; namely, this is a -bilinear map with denoting the -modulo of differential -forms on . In components . On the manifold, , the tautological tensor field, , of type is defined which assigns to each tangent space the identity linear transformation. Thus for any point and any vector , one has . In terms of the frame field, the give the expression for as , in sense that both sides yield when applied to any tangent vector in the domain of definition of the frame field. One can also consider general transformations of the linear group, , taking any basis into any other set of four linearly independent fields. The notation, , will be used below for general linear frames. Let us introduce the so-called* first deformation matrices*, and , as follows:
where and is the metric on . With this provision, we build up a* world-deformation* tensor, , which yields local tetrad deformations
and . The* first deformation matrices *, in general, are the elements of the quotient group . Here is the manifold of left cosets of the group over its subgroup . If we deform the cotetrad according to (4), we have two choices to recast metric as follows: either writing the deformation of the metric in the space of tetrads or deforming the tetrad field:
where , following [82], is called* second deformation matrix* and reads . The deformed metric splits as
where , and
The anholonomic orthonormal frame field, , relates to the tangent space metric, , by , which has the converse because . The can be decomposed in terms of symmetric, , and antisymmetric, , parts of the matrix (or, respectively, in terms of and , where ) as
where
provided is the traceless symmetric part and is the skew symmetric part of the first deformation matrix. The anholonomy objects defined on the tangent space, , read
where the anholonomy coefficients, , which represent the curls of the base members, are
In particular case of constant metric in the tetradic space, the deformed connection can be written as
The deformation comprises the following two 4D deformations and , where is the semi-Riemannian space and and are the corresponding* world-deformation* tensors. All magnitudes related to the space, , will be denoted with an over “”. According to (1), now we have and , provided
In analogy with (3), the following relations hold:
where , . So, , and
The norm of the infinitesimal displacement on the can be written in terms of the spacetime structures of as
The holonomic metric can be recast in the form
The anholonomy objects defined on the tangent space, , read
where the anholonomy coefficients, , which represent the curls of the base members, are
The connection form in terms of the frame field is then determined by
and the curvature by
where the (anholonomic) Levi-Civita (or Christoffel) connection can be written as
and is understood as the downindexed 1-form . In the usual Riemannian language involving the holonomic metric (17) defined from the tetrads, the anholonomic Christoffel connection (22) transforms into
where is the ordinary holonomic Levi-Civita connection. The corresponding curvature reduces to
provided is the ordinary Riemann tensor:
and hence, the Ricci tensor takes the form
The norm (2) can then be written in terms of the spacetime structures of and :
provided
Under a local tetrad deformation (28), a general spin connection transforms according to
We have then two choices to recast metric as follows:
In the first case, the contribution of the Christoffel symbols constructed by the metric reads
As before, the second deformation matrix, , can be decomposed in terms of symmetric, , and antisymmetric, , parts of the matrix . So,
where , is the traceless symmetric part, and is the skew symmetric part of the first deformation matrix. In analogy with (6), the deformed metric can then be split as
where
The inverse deformed metric reads
where . Hence, the usual Levi-Civita connection is related to the original connection by the relation
provided
where is the covariant derivative. The contravariant deformed metric is defined as the inverse of . Hence, the connection deformation acts like a force that deviates the test particles from the geodesic motion in the space, . A metric-affine space is defined to have a metric and a linear connection that need not be dependent on each other. In general, the lifting of the constraints of metric compatibility and symmetry yields the new geometrical property of the spacetime, which are the* nonmetricity* 1-form and the affine* torsion* 2-form representing a translational misfit (for a comprehensive discussion see [35–37, 94]). These, together with the* curvature* 2-form , symbolically can be presented as [8, 9]
where is the* covariant exterior derivative*. If the nonmetricity tensor does not vanish, the general formula for the affine connection written in the spacetime components is
where is the Riemann part and is the non-Riemann part—the affine* contortion tensor*. The torsion, given with respect to a holonomic frame, , is a third-rank tensor, antisymmetric in the first two indices, with 24 independent components. In the presence of curvature and torsion, the coupling prescription of a general field carrying an arbitrary representation of the Lorentz group will be
with denoting the corresponding Lorentz generator. The Riemann-Cartan manifold, , is a particular case of the general metric-affine manifold , restricted by the metricity condition , when a nonsymmetric linear connection is said to be metric compatible. The Lorentz and diffeomorphism invariant scalar curvature, , becomes either a function of only or :

##### 2.1. Determination of and

Let be the 1-forms of corresponding connections assuming values in the Lorentz Lie algebra. The action for gravitational field can be written in the form
where the integral,
is the usual Einstein action, with the coupling constant relating to the Newton gravitational constant , is the phenomenological action of the spin-torsion interaction, and denotes the Hodge dual (Appendix B). The variation of the connection 1-form yields
where
here we used the abbreviated notations for the wedge product monomials, , defined on the space and
The variation of the action describing the macroscopic matter sources with respect to the coframe and connection 1-form reads
where is the dual 3-form relating to the canonical energy-momentum tensor, , by
And is the dual 3-form corresponding to the canonical spin tensor, which is identical with the dynamical spin tensor , namely,
The variation of the total action, , with respect to the , , and , gives the field equations as follows:
The DC-members and can then be determined as follows:
However, it should be emphasized that the standard Riemannian (and its extensions) space interacting quantum field theory cannot be a satisfactory ground for addressing the most important processes of rearrangement of vacuum state and gauge symmetry breaking in gravity at huge energies. The difficulties here arise because the Riemannian geometry, in general, does not admit a group of isometries and it is impossible to define energy-momentum as Noether local currents related to exact symmetries. This, in turn, posed severe problem of nonuniqueness of the physical vacuum and the associated Fock space. A definition of positive frequency modes cannot, in general, be unambiguously fixed in the past and future, which leads to , because the state is unstable against decay into many particle states due to interaction processes allowed by lack of Poincaré invariance. A nontrivial Bogolubov transformation between past and future positive frequency modes implies that particles are created from the vacuum and this is one of the reasons for . So, it is another line of reasoning which supports the side suggested in this section approach. Namely, the actual advantage of our approach compared to the more conservative direct constructions, using the vierbeins, metric, and various connections, is that it allows various generalizations and alternatives of the standard gravity. In this framework, in particular, we develop an alternative* distortion gauge induced* fiber-bundle formulation of gravitation—*General Gauge Principle* (GGP) [101]. In this, we restrict ourself to consider only the simplest spacetime deformation map, (). Whereas given the principal fiber bundle with the structure group , the local coordinates are , where and , the total bundle space is a smooth manifold and the surjection is a smooth map . The collection of matter fields of arbitrary spins takes values in standard fiber over where is a set of open coverings of . The action of the structure group on defines an isomorphism of the Lie algebra of onto the Lie algebra of vertical vector fields on tangent to the fiber at each called fundamental. Note that an invariance of the Lagrangian under spacetime diffeomorphisms in implies an invariance of under the group and vice versa if, and only if, the generalized local gauge transformations of the fields and their covariant derivative are introduced by finite local gauge transformations:
Here denotes the covariant derivative agreed with the metric . For example, for the spinor field , where are the Dirac matrices, , , and . To involve a drastic revision of the role of gauge fields in the physical concept of the spacetime deformation/distortion, we generalize the standard gauge scheme by exploring a new special type of* distortion* gauge field, , which is assumed to act on the external spacetime groups. That is, we also consider the principle fiber bundle, , with the base space , the structure group , and the surjection . The matter fields take values in the standard fiber which is the Hilbert vector space where a linear representation of group is given. This space can be regarded as the Lie algebra of the group upon which the Lie algebra acts according to the law of the adjoint representation: . The GGP framework accounts for the* gravitation gauge group * generated by the hidden local internal symmetry . We pursue a principle goal to build up the* world-deformation* tensor, , where is the differential form of gauge field . We connect the structure group , further, to the nonlinear realization of the Lie group of* distortion* of the . The nonlinear realization technique or the method of phenomenological Lagrangians [104–110] provides a way to determine the transformation properties of fields defined on the quotient space. In accord, we treat the distortion group and its stationary subgroup , respectively, as the dynamical group and its algebraic subgroup. The fundamental field is distortion gauge field (a) and, thus, all the fundamental gravitational structures in fact, the metric as much as the coframes and connections, acquire a* distortion-gauge induced* theoretical interpretation. We derive the Maurer-Cartan structure equations, where the distortion fields (a) are treated as the Goldstone fields. This framework explores the most important processes of rearrangement of vacuum state and a spontaneous breaking of gravitation gauge symmetry at huge energies.

##### 2.2. TSSD: Revisited

For the benefit of the reader, a brief outline of the key ideas behind the TSSD [82], as a guiding principle, is given in this section to make the rest of the paper understandable. Before we report on the physical foundation of post-Riemannian geometry, we may remark on the form of generic spacetime deformation, . When torsion is nonvanishing, the affine connection is no longer coincident with the Levi-Civita connection, and the geometry is no longer Riemannian, but one has a RC spacetime, , with a nonsymmetric, but metric-compatible, connection. Teleparallel gravity, in turn, represented a new way of including torsion into general relativity, an alternative to the scheme provided by the usual Einstein-Cartan-Sciama-Kibble approach. However, the gravitational coupling of the fundamental fields in teleparallel gravity is a very controversial subject [15, 111–114]. The basic difficulty lies in the definition of the spin connection and consequently in the correct form of the gravitational coupling prescription. For a specific choice of the free parameters, teleparallel gravity becomes completely equivalent to GR. In this case it is usually referred to as the teleparallel equivalent of GR. From this point of view, curvature and torsion are simply alternative ways of describing the gravitational field and are consequently related to the same degrees of freedom of gravity. Teleparallel gravity attributes gravitation to torsion, but in this case torsion accounts for gravitation not by geometrizing the interaction, but by acting as a force. The fundamental difference between these two theories above was that, whereas in the former a torsion is a propagating field having as a source—the energy-momentum tensor, in the latter a torsion is not a propagating field having as a source—the total spin, a point which can be considered a drawback of this model. In fact, the two physical interpretations of torsion described above are clearly conflictive. This problem can be solved just only by experiment. Therefore, we have to separate, from the very outset, these two completely different cases. This reasoning supports our choice of a double deformation map. Namely, following [82], we assume that the spacetime deformation comprises the two ingredient deformations . Hence, local tetrad deformations (13) are performed according to the following heuristic map, in* two steps* (two-step deformation map):