Modified Theories of Gravitation behind the Spacetime Deformation

Let us consider a smooth deformation map $$, written in terms of the world-deformation tensor $$, general, $$, and flat, M₄, smooth differential 4D-manifolds. The tensor, $$, can be written in the form $$, where the distortion-complex (DC) includes the invertible distortion matrix $$ and the tensor $$ and ∂_l : = ∂/∂x^l). Here we use Greek alphabet (μ, ν, ρ, … = 0,1, 2,3) to denote the holonomic world indices related to $$ and the second half of Latin alphabet (l, m, k, … = 0,1, 2,3) to stand for the world indices related to M₄. The principle foundation of a world-deformation tensor comprises the following two steps [101]: (1) the basis vectors e_m at given point (p ∈ M₄) undergo distortion transformations by means of the matrix $$; and (2) the diffeomorphism $$ is constructed by seeking a new holonomic coordinates x^μ(x^l) 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 ds of infinitesimal displacement dx^μ on $$ in terms of the spacetime structures of M₄: The holonomic metric on the space $$ can be recast in the form g = g_μνϑ^μ ⊗ ϑ^ν = g(e_μ, e_ν)ϑ^μ ⊗ ϑ^ν, with the components g_μν = g(e_μ, e_ν) in dual holonomic basis {ϑ^μ ≡ dx^μ}. 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, e, as a shorthand for the collection of the 4-tuplet (e₀, …, e₃), where $$. We use the first half of Latin alphabet (a, b, c, … = 0,1, 2,3) to denote the anholonomic indices related to the tangent space. The frame field, e, then defines a dual vector, ϑ, of differential forms, $$, 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 C^∞-bilinear map ⌋:Ω¹ → Ω⁰ with Ω^p denoting the C^∞-modulo of differential p-forms on $$. In components $$. On the manifold, $$, the tautological tensor field, id, of type (1,1) is defined which assigns to each tangent space the identity linear transformation. Thus for any point $$ and any vector $$, one has id(ξ) = ξ. In terms of the frame field, the ϑ^a give the expression for id as id : = eϑ = e₀ ⊗ ϑ⁰ + ⋯e₃ ⊗ ϑ³, in sense that both sides yield ξ when applied to any tangent vector ξ in the domain of definition of the frame field. One can also consider general transformations of the linear group, GL(4, R), taking any basis into any other set of four linearly independent fields. The notation, {e_a, ϑ^b}, will be used below for general linear frames. Let us introduce the so-called first deformation matrices, $$ and $$, as follows: where $$ and η_ks is the metric on M₄. With this provision, we build up a world-deformation tensor, $$, which yields local tetrad deformations and $$. The first deformation matrices π, in general, are the elements of the quotient group $$. Here $$ is the manifold of left cosets of the group $$ over its subgroup SO(3,1). If we deform the cotetrad according to (4), we have two choices to recast metric as follows: either writing the deformation of the metric in the space of tetrads or deforming the tetrad field: where γ_lm, following [82], is called second deformation matrix and reads $$. The deformed metric splits as where $$, and The anholonomic orthonormal frame field, e, relates g to the tangent space metric, o_ab = diag(+−− −), by $$, which has the converse $$ because $$. The γ_lm can be decomposed in terms of symmetric, π_(al), and antisymmetric, π_[al], parts of the matrix $$ (or, respectively, in terms of $$ and $$, where $$) as where provided Θ_al is the traceless symmetric part and φ_al 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 $$, where V₄ is the semi-Riemannian space and $$ and Ω are the corresponding world-deformation tensors. All magnitudes related to the space, V₄, 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 V₄ can be written in terms of the spacetime structures of M₄ 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 $$ is the ordinary Riemann tensor: and hence, the Ricci tensor takes the form The norm id  (2) can then be written in terms of the spacetime structures of V₄ and M₄: provided Under a local tetrad deformation (28), a general spin connection transforms according to We have then two choices to recast metric as follows: In the first case, the contribution of the Christoffel symbols constructed by the metric $$ reads As before, the second deformation matrix, γ_ab, can be decomposed in terms of symmetric, π_(ab), and antisymmetric, π_[ab], parts of the matrix $$. So, where $$, Θ_ab is the traceless symmetric part, and φ_ab 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 g^νρ 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, V₄. A metric-affine space $$ is defined to have a metric and a linear connection that need not be dependent on each other. In general, the lifting of the constraints of metric compatibility and symmetry yields the new geometrical property of the spacetime, which are the nonmetricity 1-form N_ab and the affine torsion 2-form T^a representing a translational misfit (for a comprehensive discussion see [35–37, 94]). These, together with the curvature 2-form $$, 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, $$ given with respect to a holonomic frame, dϑ^ρ = 0, is a third-rank tensor, antisymmetric in the first two indices, with 24 independent components. In the presence of curvature and torsion, the coupling prescription of a general field carrying an arbitrary representation of the Lorentz group will be with J_ab denoting the corresponding Lorentz generator. The Riemann-Cartan manifold, U₄, is a particular case of the general metric-affine manifold $$, restricted by the metricity condition N_λμν = 0, when a nonsymmetric linear connection is said to be metric compatible. The Lorentz and diffeomorphism invariant scalar curvature, R, becomes either a function of $$ only or g_μν:

At low energies the spacetime group associated with the matter fields is the Poincaré group. An extension of the Poincaré gauge theory of gravity constructed in the RC geometry, to the most general spacetime symmetry gauge theory, the MAG theory, has been developed by Hehl and collaborators [17, 18]. The MAG theory has the most general type of covariant derivative: in addition to curvature and torsion, the MAG also has nonmetricity, that is, a nonmetric compatible connection. Hence parallel transport no longer preserves length and angle. Note that there are only indications (but no conclusive evidence) for assuming invariance of physical systems under the action of the entire affine group and that in MAG one is far from actually calculating S-matrix elements. Although the theoretical structure of this theory has been developed, we do not yet have much understanding of what new physics are allowed by the MAG theory. One source of improved understanding is exact solutions; see, for example, [116–118] and references therein. But due to the highly nonlinear nature of theory, exact solutions are not easily found unless they have a great deal of symmetry. However, to carry in full generality through the extension of TSSD-ideas as applied to more general metric-affine gravity, it reasonable as a next step to gauge immediately the 4 + 16 parameter affine group A(4, R) = R⁴  GL(4, R), which lacks a metric structure altogether, and to introduce the metric subsequently. General affine invariance adds dilation and shear invariance as physical symmetries to Poincaré invariance, and both of these symmetries are of physical importance. Dilation invariance is a crucial component of particle physics in the high energy regime. Shear invariance was shown to yield representations of hadronic matter; the corresponding shear current can be related to hadronic quadrupole excitations. From this in the framework of the gauge theory of the affine group with a metric supplemented, as a physically meaningful field theory, it is speculated that the invariance under affine transformations played an essential part at an early stage of the universe, such that todays Poincaré invariance might be a remnant of affine invariance after some symmetry breaking mechanism. Thus MAG encompasses the PG as a subcase. The rather comprehensive gauging of the affine group was done in [17, 18], where a rigorous mathematical justification of the relevant spacetime structures and equations with more details is to be found. But a brief outline of the key points of relevance to the context of TSSD can be stated here, in which some conventions and results will be borrowed from the presentation given in [17, 18, 42]. To this aim, one enlarges at any point of the base manifold $$ a tangent space $$ to an affine tangent space $$ by allowing freely translating elements of $$ to different points $$. The collection of all affine tangent spaces $$ forms the affine bundle $$. An affine frame of $$ at x is a pair $$ (whereas before A = π, σ) consisting of a linear frame $$ and a point $$. The origin of $$ is that point $$ for which the affine frame $$ reduces to the linear frame $$. The transformation behavior of an affine frame $$ under an affine transformation (Λ, τ) with τ = τ^a ∈ T⁴≃R⁴ and $$ reads The affine group acts transitively on the affine tangent spaces $$: any two affine frames of some $$ can be related by a unique affine transformation. Consequently, the notion of an affine frame should be enlarged to include all GL(4, R)-representations needed. The gauging is accomplished by the introduction of the generalized affine connection as a prescription $$, which maps infinitesimally neighbouring affine tangent spaces $$, $$, where $$, by an A(4, R)-transformation onto each other. The generalized affine connection consists of a GL(4, R)-valued 1-form $$ and an R⁴-valued one-form $$, both of which generate the required A(4, R)-transformation. The two affine tangent spaces get now related by an affine transformation according to the prescription Introducing origins in $$, that is, by soldering $$ to $$, one losts translational invariance in $$ but gained a local one-to-one correspondence between translations in $$ and diffeomorphisms on $$. However, one can introduce translational invariance by demanding diffeomorphism invariance instead and continue to work with this modified notion of translation invariance. The diffeomorphisms themself, as horizontal transformations in their active interpretation, cannot be gauged according to the usual gauge principle and thus do not furnish their own gauge potential. In the Lagrangian formulation of a general metric-affine theory it was assumed that the matter fields are described in terms of manifields. Actually, the matter fields are supposed to be represented by vector- or spinor-valued p-forms. However, in MAG one goes beyond Poincaré invariance, assuming that matter fields might undergo not only Poincaré transformations but also the more general linear transformations. In this case, the unavailability of local Lorentz frames poses no problem in the context of boson fields. They are naturally constructed so as to be capable of carrying the action of SL(4, R), instead of the Lorentz group, whether in a local frame or holonomically. However, this is not true of the conventional fermion fields one uses to represent matter, and a linear action should nevertheless be realized, through the use of infinite-component linear field representations of the double covering of the linear, affine, and diffeomorphism groups. More general spinor representations than the Poincaré-representations have to be constructed. Such representations of the matter fields corresponding to the affine group must exist. Otherwise it does not make sense to demand A(4, R)-invariance of a nonvacuum field theory. The construction of the spinor representations of fermionic matter fields in MAG, the so-called manifields, is illustrated in [17, 18]. These representations turn out to be infinite dimensional, due to the noncompactness of the gauge (sub)group GL(4, R). Fermions are then assigned to spinor manifields. The restriction of GL(4, R) to SO(1, n − 1) reduces the manifield representations to the familiar spinor representations. The actual gauging of the affine group introduced, in addition to the matter fields f, the gravitational gauge potentials $$ and $$. Following the common practice, we will use as gauge potential the translation invariant $$ in place of the translational part $$ of the affine connection, simply because it has the immediate interpretation as a reference (co)frame. Expanded in a holonomic frame, the components of $$ and $$ differ just by a Kronecker symbol, as is clear from the definition of $$. Also the homogeneous transformation behavior of $$ will turn out to be quite convenient. For the action of the GL(4, R)-gauge potential, below the shorthand notation $$ is used instead of $$. The introduction of a metric into MAG is mandatory since we are interested in a realistic macroscopic gravity theory that contains GR in some limit. So, it was assumed as a metric of the general form $$ with coefficients $$ which are independent of the coframe $$. The gauge potentials $$ become true dynamical variables if one has to add to the minimally coupled matter Lagrangian $$ a gauge Lagrangian V. Here, we restrict ourself considering only first order Lagrangian V, which must be expressed in terms of the gauge potentials and their first derivatives. The total action reads then If Ψ, as a p-form, represents a matter field (fundamentally a representation of the SL(4, R) or of some of its subgroups), its first order Lagrangian $$ will be embedded in metric-affine spacetime by the minimal coupling procedure, that is, exterior covariant derivatives feature in the kinetic terms of the Lagrangian instead of only exterior ones. Just as ordinary stress is the analogue of the (Hilbert) energy-momentum density, in MAG, one has, in addition, the spin current and the dilation plus shear currents inducing the torsion and nonmetricity fields, respectively. Both spin and dilation plus shear are components of the hypermomentum current, symmetric for dilation plus shear and antisymmetric for spin: spin current ⊗ dilation current ⊗ shear current. And these currents ought to couple to the corresponding post-Riemannian structures. In accord, the material currents are defined as follows: where $$ is the metric (and symmetric) energy-momentum current of matter (Hilbert current), whereas one believes that $$ should be the dual 3-form relating to the canonical energy-momentum tensor, $$, by where we used the abbreviated notations for the wedge product monomials, $$, and ⋆ denotes the Hodge dual (Appendix B). The canonical energy-momentum density is identical with the dynamical tetrad energy-momentum density $$, where $$, and the canonical tensor $$ is generally not symmetric. The relation between the tetrad dynamical energy-momentum tensor and the metric dynamical energy-momentum tensor for matter fields is $$. The canonical hypermomentum current $$, which couples to the linear connection, can be decomposed according to with $$ as (dynamical) spin current, $$, as dilation current, and $$ as symmetric and tracefree shear current, which is a bit more remote from direct observation than the other currents. From variation of a total action (73), we find the matter and the gauge field equations as follows: where we take into account that the variation of $$ with respect to the affine connection in MAG is equivalent to the variation with respect to the torsion (or contortion) or spin connection: ($$). Note that, using the Noether identities, the zeroth field equation can be shown to be redundant, provided the matter equations hold. In first, the canonical energy-momentum of the translational gauge potential $$ can be written in standard form Thus, the equations of the standard MAG theory written in the framework of the first order Lagrangian, which is expressed in terms of the gauge potentials and their first derivatives, can be recovered for A = π: Consequently, the (2nd) equation defines a nondynamical torsion, such that torsion at a given point in spacetime does not vanish only if there is matter at this point, represented in the Lagrangian density by a function which depends on torsion. Unlike the metric, which is related to matter through a differential field equation, torsion does not propagate. However, equations (79) can be equivalently replaced by the set of modified MAG equations for A = σ: where in which the torsion is dynamical if only $$. Actually, in testing the general MAG equations (77), we desire, in some limit, to recover the field equation for different (sub)cases; then we have to put on Lagrange multipliers, whereas, in the case of TSSD-PG, one has to kill nonmetricity; TSSD-EC is the TSSD-PG with the curvature scalar as gravitational Lagrangian; in the case of TSSD-teleparallel gravity (TSSD-GR_||) in a Weitzenböck spacetime, one has to remove nonmetricity and curvature; and, finally, in the case of GR in a Riemannian space, curvature scalar as Lagrangian, one has to remove nonmetricity and torsion. To recover, for example, GR, let us in (68) choose $$ as the torsion of the connection $$; that is, $$. In this case, $$, and we are left with the torsionless, a SO(3,1) valued Lorentz spin connection of general relativity: $$. Then, one can choose for the missing piece of the gauge field Lagrangian V_GR the corresponding 4-form of the curvature scalar [8, 9] and take Lagrange multipliers for extinguishing nonmetricity and torsion: where æ is the coupling constant relating to the Newton gravitational constant æ = 8πG/c⁴, $$ is the curvature tensor of the independent field variable $$, and $$ is the eta basis consisting in the Hodge dual of exterior products of tetrads by means of the Levi-Civita object (Appendix B).

The RC manifold, U₄, is a particular case of general metric-affine manifold M₄, restricted by the metricity condition, when a nonsymmetric linear connection is said to be metric compatible. Taking the antisymmetrized derivative of the metric condition gives an identity between the curvature of the spin connection and the curvature of the Christoffel connection where we take into account (65), so Hence, the relations between the scalar curvatures for the U₄ manifold read This means that the Lorentz and diffeomorphism invariant scalar curvature, R, becomes either a function of $$ only or a function of g_μν only. Certainly, it can be seen by noting that the Lorentz gauge transformations can be used to fix the six antisymmetric components of $$ to vanish. Then in both cases diffeomorphism invariance fixes four more components out of the six g_μν, with the four components g_0μ being nondynamical, obviously, leaving only two dynamical degrees of freedom. This shows the equivalence of the vierbein and metric formulations holds. To recover the TSSD-U₄ theory, one can choose for the missing piece the EC Lagrangian V_EC (here without cosmological constant): provided $$ is equal to $$. The variation of the total action (73), given by the sum of the gravitational field action S_EC = ∫V_EC with the Lagrangian (86) and the macroscopic matter sources $$, with respect to the $$, 1-form $$ and Ψ, gives where $$. The metricity condition $$ yields $$, and hence $$. Consequently, the field equations (87) become where $$ is the dual 3-form corresponding to the canonical spin tensor, which is identical with the dynamical spin tensor $$; namely, provided and that According to (85), the relations between the Ricci scalars read To obtain some feeling for the tensor language then we may recast the first equation in (88) into the form Making use of the relation, gives We may evaluate the second equation in (88) as Taking into account the relation we then obtain It is remarkable that the torsion (98), in a natural way, can be made a short-range propagating torsion. Actually, from the above equations, we see that it is the spin $$ and spacetime deformations π(x) and σ(x) that define the torsion $$: which through the field equation, in turn, defines Einstein’s field tensor $$. A generic spacetime deformation, π(x), consists of two ingredient deformations $$ of the orthonormal frame. While the deformation matrix $$ implies a peculiar condition (53), the choice of the σ(x) is not fixed yet. This allows us to impose a physical constraint upon the spacetime deformation σ(x): where □ is a generalization of the d’Alembertian operator for covariant derivatives defined on the RC manifold, U₄. Then, the modified EC equations (95) and (98) reduce to which describe the short-range propagating torsion and spin-spin interaction. At large distances r > λ_T ≡ ℏ/M_Tc (Compton length), torsion vanishes $$. Note that the short-range propagating torsion is of fundamental importance from a view point of microphysics.

We may impose some other physical constraints upon the spacetime deformation σ(x), which will be useful for the theory of electromagnetism and charged particles: with φ as a scalar or pseudoscalar function of relevant variables. Then which recovers the term in the Lagrangian of pseudoscalar-photon interaction theory [95–100], such that the nonmetric part of the Lagrangian can be put in the well known form of the χ − g framework: where F_μν = A_μ,ν − A_ν,μ have the usual meaning for electromagnetism. This is equivalent, up to integration by parts in the action integral (modulo a divergence), to the Lagrangian According to (105), the gravitational constitutive tensor χ^σνμρ = χ^μρσν = −χ^μρνσ [119] of the gravitational fields (e.g., metric g_μν, (pseudo) scalar field φ, etc.) reads Specifying the physical meaning of the constrains (99), (101), and (102), will be an interesting topic for another investigation elsewhere. Here we finally concentrate on the other (sub)case when $$ and the connection $$ vanishes, which characterizes teleparallel gravity. In this case, the resulting connection has the form $$ where $$ is the contortion tensor written in terms of the Weitzenböck torsion $$. The particle equation of motion then becomes the force equation of teleparallel gravity. Hence, the Weitzenböck covariant derivative of the tetrad field vanishes identically: $$. This is the so-called distant, or absolute parallelism condition. As a consequence of this condition, the corresponding Weitzenböck spin connection also vanishes identically: $$. Of course, these relations above are true only in one specific class of frames. In fact, since $$ is the Weitzenböck spin connection, if it vanishes in a given frame, it will be different from zero in any other frame related to the first by a local Lorentz transformation. The teleparallel gravity becomes consistent and fully equivalent with GR, even in the presence of spinor fields if we write the minimal coupling prescription as $$ with $$ the teleparallel Fock-Ivanenko derivative written in the form $$, where the teleparallel spin connection, $$, reads $$. Field equations can be derived from the least action, $$, with the Lagrangian of teleparallel gravity, where we take Lagrange multipliers for extinguishing nonmetricity and curvature. This holds when the deformation σ(x) vanishes identically (σ(x) ≡ 0): where the three irreducible pieces of the torsion $$ under SO(1,3)-decomposition are The projective transformation with arbitrary 1-form field P, leaves the Hilbert-Einstein type Lagrangian invariant, which leads to the connection determined up to a 1-form. Projectively related connections have the same (unparametrized) geodesics. So, only projectively invariant matter Lagrangians would be allowed. This necessitates abandoning this constraint, at the very least replacing semi-Riemannian geometry by Weyl’s. Namely, to remove this constraint from the gravitational Lagrangians (82) and (86), following [120, 121], we may lift the Lagrange multiplier λ^ab and add a dilaton type massless scalar field to in the context of the Weyl 1-form $$, which is of the type of a gauge potential for dilations anyways: respectively, provided the trace $$ of a connection is closely related to the Weyl 1-form [17, 18], as In case if matter is present and supplies energy-momentum and hypermomentum currents, then the hypermomentum, via the second field equation (77), turns out to be proportional to the post-Riemannian pieces of the connection.

In case of translational gauge with the most general term $$, quadratic in$$ the field equation $$ then becomes where $$ is linear in $$, and the energy-momentum current of the gauge field. This can be considered as a starting point for turning to the Lagrangians with the quadratic ansatz for the kinetic term, that is, quadratic in the field strengths. Note that recently the authors of [122] show that the inverse of the so-called Barbero-Immirzi parameter multiplying the pseudoscalar curvature, because of the topological Nieh-Yan form, can only be appropriately discussed if torsion square pieces are included. Using more conservative direct constructions, they establish the exact relations between both approaches by applying the topological Euler and Pontryagin forms in a Riemann-Cartan space expressed for the first time in terms of irreducible pieces of the curvature tensor. Only in a Riemann-Cartan spacetime, that is, in a spacetime with torsion, parity violating terms can be brought into the gravitational Lagrangian in a straightforward and natural way. They conclude that Riemann-Cartan spacetime is a natural habitat for chiral fermionic matter fields. However, the standard EC theory is invariant under the spacetime diffeomorphisms and local SO(3,1) transformations acting in the tangent space. In our approach we suggest extending the SO(3,1) to local GL(4, R) group. This group is surely not the invariance of the action of standard theory, and so it takes the latter into diverse other ones, still invariant under spacetime diffeomorphisms.