Invariant Field Equations Research Articles

Action Principle for Scale Invariance and Applications (Part I)

On the basis of a general action principle, we revisit the scale invariant field equation using the cotensor relations by Dirac (1973). This action principle also leads to an expression for the scale factor λ, which corresponds to the one derived from the gauging condition, which assumes that a macroscopic empty space is scale-invariant, homogeneous, and isotropic. These results strengthen the basis of the scale-invariant vacuum (SIV) paradigm. From the field and geodesic equations, we derive, in current time units (years, seconds), the Newton-like equation, the equations of the two-body problem, and its secular variations. In a two-body system, orbits very slightly expand, while the orbital velocity keeps constant during expansion. Interestingly enough, Kepler’s third law is a remarkable scale-invariant property.

Solitons in several space dimensions with variable exponents

The authors examine the Lorentz invariant nonlinear field equations with a variable exponent in several space dimensions in order to prove the existence of soliton‐like solutions. The existence of a static solution minimizing the energy is proved. Under some addition assumptions, the existence of infinitely many solutions is also proved.

The Weyl double copy in vacuum spacetimes with a cosmological constant

We examine the Weyl double copy relation for vacuum solutions of the Einstein equations with a cosmological constant using the approach we previously described, in which the spin-1/2 massless free-field spinors (Dirac-Weyl fields) are regarded as basic units. Based on the exact non-twisting vacuum type N and vacuum type D solutions, the finding explicitly shows that the single and zeroth copies fulfill conformally invariant field equations in conformally flat spacetime. In addition, irrespective of the presence of a cosmological constant, we demonstrate that the zeroth copy connects Dirac-Weyl fields with the degenerate electromagnetic fields in the curved spacetime in addition to connecting gravity fields with the single copy in conformally flat spacetime. Moreover, the study also demonstrates the critical significance the zeroth copy plays in time-dependent radiation solutions. In particular, for Robinson-Trautman (Λ) gravitational waves, unlike the single copy, we find that the zeroth copy carries additional information to specify whether the sources of associated gravitational waves are time-like, null, or space-like, at least in the weak field limit.

Conjugate spinor field equation for massless spin-$\frac{3}{2}$ field in de Sitter ambient space

The quantum field theory in de Sitter ambient space provide us with a comprehensivedescription of massless gravitational field.Using the gauge-covariant derivative in the de Sitter ambient space, the gauge invariant Lagrangian density has been found.In this paper, the equation of the conjugate spinor for massless spin-$\frac{3}{2}$ field is obtained by Euler-Lagrange equation. Then the field equation is written in terms of the Casimir operator of the de Sitter group. Finally, the gauge invariant field equation is presented.

De Sitter field equations from quadratic curvature gravity: A group theoretical approach

In this paper, the linearized field equations related to the quadratic curvature gravity theory have been obtained in the four-dimensional de Sitter (dS) space–time. The massless spin-2 field equations have been written in terms of the Casimir operators of dS group making use of the ambient space notations. By imposing some simple constraints, arisen from group theoretical interpretation of the field equations, a new four-dimensional Gauss–Bonnet (GB-)like action has been introduced with the related field equations transforming according to the unitary irreducible representations (UIRs) of dS group. Since, the field equations transform according to the UIRs of dS group, the GB-like action, we just obtained, is expected to be a successful model of modified gravity. For more clarity, the gauge invariant field equations have been solved in terms of a gauge-fixing parameter [Formula: see text]. It has been shown that the solution can be written as the multiplication of a symmetric rank-2 polarization tensor and a massless minimally coupled scalar field on dS space. The Krein–Gupta–Bleuler quantization method has been utilized and the covariant two-point function has been calculated in terms of the massless minimally coupled scalar two-point function, using the ambient space notations. It has been written in terms of dS intrinsic coordinates from the ambient space counterpart. The two-point functions are dS invariant and free of any theoretical problems. It means that the proposed model is a successful model of modified gravity and it can produce significant results in the contexts of classical theory of gravity and quantum gravity toy models.

Construction of some classes of nonlinear PDE’s admitting soliton solutions

We study a class of Lorentz invariant nonlinear field equations in multiple space dimensions. The main purpose is to obtain soliton-like solutions with variable exponent. The fields are characterized by a topological invariant, which we call the charge. W

Gauge theory of massless spin- field in de Sitter space-time

On several levels of theoretical physics, especially particle physics and early universe cosmology, de Sitter space-time has become an attractive possibility. The principle of local gauge invariance governs all known fundamental interactions of elementary particles, from electromagnetism and weak interactions to strong interactions and gravity. This paper presents a procedure for defining the gauge-covariant derivative and gauge invariant Lagrangian density in de Sitter ambient space-time formalism. The gauge invariant field equation is then explicitly calculated in detail for a massless spin- gauge field.

Higher-rank fields and currents

Sp(2M) invariant field equations in the space ℳ M with symmetric matrix coordinates are classified. Analogous results are obtained for Minkowski-like subspaces of ℳ M which include usual 4d Minkowski space as a particular case. The constructed equations are associated with the tensor products of the Fock (singleton) representation of Sp(2M) of any rank r. The infinite set of higher-spin conserved currents multilinear in rank-one fields in ℳ M is found. The associated conserved charges are supported by $$ \mathrm{r}M-\frac{\mathrm{r}\left(\mathrm{r}-1\right)}{2} $$ -dimensional differential forms in ℳ M , that are closed by virtue of the rank-2r field equations. The cohomology groups H p (σ − r ) with all p and r, which determine the form of appropriate gauge fields and their field equations, are found both for ℳ M and for its Minkowski-like subspace.

Metric measure space as a framework for gravitation

In this manuscript, we show how conformal invariance can be incorporated in a classical theory of gravitation, in the context of metric measure space. Metric measure space involves a geometrical scalar f, dubbed as density function, which here appears as a conformal degree of freedom. In this framework, we present conformally invariant field equations, the relevant identities and geodesic equations. In metric measure space, the volume element and accordingly the operators with integral based definitions are modified. For instance, the divergence operator in this space differs from the Riemannian one. As a result, a gravitational theory formulated in this space has a generalized second Bianchi identity and a generalized conservation of energy-momentum tensor. It is shown how, by using the generalized identity for conservation of energy-momentum tensor, one can obtain a conformally invariant geodesic equation. By comparison of the geodesic equations in metric measure space with the Bohmian trajectories, in both relativistic and non-relativistic regimes, a relation between density function f and the quantum potential is proposed. This suggests metric measure space to be considered as a suitable framework for geometric description of Bohm’s quantum mechanics. On the other hand, as it is known, Weyl geometry is one of the main approaches to construct conformally invariant gravitational models. Regarding the fact that the connection in the integrable Weyl space is modified and in metric measure space remains the same as it is in the Riemann space, the mathematical analogy between these two spaces is also discussed.

Scalar perturbation potentials in a homogeneous and isotropic Weitzenböck geometry

We describe the fully gauge invariant cosmological perturbation equations in teleparallel gravity by using the gauge covariant version of the Stewart lemma for obtaining the variations in tetrad perturbations. In teleparallel theory, perturbations are the result of small fluctuations in the tetrad field. The tetrad transforms as a vector in both its holonomic and anholonomic indices. As a result, in the gauge invariant formalism, physical degrees of freedom are those combinations of perturbation parameters which remain invariant under a diffeomorphism in the coordinate frame, followed by an arbitrary rotation of the local inertial (Lorentz) frame. We derive these gauge invariant perturbation potentials for scalar perturbations and present the gauge invariant field equations governing their evolution.

Onp-form theories with gauge invariant second order field equations

We explore field theories of a single p-form with equations of motions of order strictly equal to two and gauge invariance. We give a general method for the classification of such theories which are extensions to the p-forms of the Galileon models for scalars. Our classification scheme allows to compute an upper bound on the number of different such theories depending on p and on the space-time dimension. We are also able to build a non trivial Galileon like theory for a 3-form with gauge invariance and an action which is polynomial into the derivatives of the form. This theory has gauge invariant field equations but an action which is not, like a Chern-Simons theory. Hence the recently discovered no-go theorem stating that there are no non trivial gauge invariant vector Galileons (which we are also able here to confirm with our method) does not extend to other odd p cases.

Conformally covariant vector–spinor field in de Sitter space

In this paper, we study conformally invariant field equations for a vector–spinor \(\left( \mathrm{spin-}\frac{3}{2}\right) \) field in the de Sitter space-time. The solutions are also obtained in terms of the de Sitter–Dirac plane waves. The related two-point functions are calculated in both the de Sitter ambient space formalism and intrinsic coordinates. In order to study the conformal invariance, Dirac’s six-cone formalism is utilized in which the field equations are expressed in a manifestly conformal way in \(4+2\)-dimensional conformal space and then followed by a projection to the de Sitter space.

On the equivalence of the FRW field equations and those of Newtonian cosmology

We present a simple argument to explain why the field equations of the Friedmann-Robertson-Walker metric are equivalent to those of Newtonian cosmology. By passing to the infinite limit of a family of conformally rescaled FRW metrics in suitable coordinates, we reveal Newtonian space and time. The limiting process preserves the Einstein equations and these may be elucidated directly from the Newtonian limit up to the determination of the scalar curvature parameter. Consideration of the conformally invariant scalar field equation on the FRW spacetime is used to recover the Einstein equations efficiently from the Newtonian theory. We proceed to examine the limiting procedure in connection with the Cartan formulation of Newtonian gravity.

Conformally and gauge invariant spin-2 field equations

Using an approach based on the Casimir operators of the de Sitter group, the conformal invariant equations for a fundamental spin-2 field are obtained, and their consistency discussed. It is shown that, only when the spin-2 field is interpreted as a 1-form assuming values in the Lie algebra of the translation group, rather than a symmetric second-rank tensor, the field equation is both conformal and gauge invariant.

Constructing conformally invariant equations using the Weyl geometry

We present a simple, systematic and practical method to construct conformally invariant equations in arbitrary Riemann spaces. This method that we call ‘Weyl-to-Riemann’ is based on two features of the Weyl geometry. (i) Weyl space is defined by the metric tensor and the Weyl vector W; it is equivalent to the Riemann space when W is a gradient. (ii) Any homogeneous differential equation written in the Weyl space by means of the Weyl connection is conformally invariant. The Weyl-to-Riemann method selects those equations whose conformal invariance is preserved when reducing to the Riemann space. Applications to scalar, vector and spin-2 fields are presented, which demonstrate the efficiency of this method. In particular, a new conformally invariant spin-2 field equation is exhibited. This equation extends Grishchuk–Yudin's equation and fixes its limitations since it does not require the Lorenz gauge. Moreover, this equation reduces to the Drew–Gegenberg and Deser–Nepomechie equations in Minkowski and de Sitter spaces, respectively.

A NEW PHYSICAL STATE FOR DE SITTER LINEAR GRAVITY

Based on conformal invariance and using Dirac's six-cone formalism, a new conformally invariant physical field equation for de Sitter (dS) linear gravity has been obtained, which corresponds to one of the unitary irreducible representations of the dS group and is denoted by [Formula: see text] in the sense of discrete series. Using ambient space notations, it has been shown that the solution to this new field equation can be written as the multiplication of a generalized symmetric polarization tensor of rank 2 and a massless conformally coupled scalar field in dS space–time. The physical tensor two-point function has been calculated in terms of the conformally coupled scalar two-point function in the ambient space formalism. It has been expressed in terms of dS intrinsic coordinates from its ambient space counterpart, which is dS-invariant and free of any divergences.

Auxiliary “Massless” Spin-2 Field in De Sitter Universe

For the tensor field of rank-2 there are two unitary irreducible representation (UIR) in de Sitter (dS) space denoted by $\Pi^{\pm}_{2,2}$ and $\Pi^{\pm}_{2,1}$ [1]. In the flat limit only the $\Pi^{\pm}_{2,2}$ coincides to the UIR of Poincar\'e group, the second one becomes important in the study of conformal gravity. In the pervious work, Dirac's six-cone formalism has been utilized to obtain conformally invariant (CI) field equation for the "massless" spin-2 field in dS space [2]. This equation results in a field which transformed according to $\Pi^{\pm}_{2,1}$, we name this field the auxiliary field. In this paper this auxiliary field is considered and also related two-point function is calculated as a product of a polarization tensor and "massless" conformally coupled scalar field. This two-point function is de Sitter invariant.

Radiation fields of a uniformly accelerating point source in the framework of Stueckelberg’s manifestly covariant relativistic dynamics

In this paper we explore the problem of fields generated by a source undergoing hyperbolic motion in the framework of Stueckelberg manifestly covariant relativistic dynamics. The resulting gauge fields are computed numerically using Green-Functions which are retarded in the Stueckelberg absolute time τ, and qualitatively compared with Maxwell fields generated by the same motion. The gauge invariant field equations are second order in this parameter as well as in space-time, resulting in identification of a five-dimensional manifold for the gauge fields. We find that although the zero mode of all fields coincides with the corresponding Maxwell fields, the generalized Lorentz force (necessarily involving a fifth component of the gauge field) depends on the nonzero modes as well, which affects the motion of particles subject to these forces.

Gauge fields in (A)dS d within the unfolded approach: algebraic aspects

It has recently been shown that generalized connections of the (A)dS space symmetry algebra provide an effective geometric and algebraic framework for all types of gauge fields in (A)dS, both for massless and partially-massless. The equations of motion are equipped with a nilpotent operator called $\sigma_-$ whose cohomology groups correspond to the dynamically relevant quantities like differential gauge parameters, dynamical fields, gauge invariant field equations, Bianchi identities etc. In the paper the $\sigma_-$-cohomology is computed for all gauge theories of this type and the field-theoretical interpretation is discussed. In the simplest cases the $\sigma_-$-cohomology is equivalent to the ordinary Lie algebra cohomology.

On conformal, [formula omitted] and [formula omitted] symmetries of massless fields

The sp ( 8 , R ) invariant formulation of free field equations of massless fields of all spins in AdS 4 available previously in terms of gauge invariant field strengths is extended to gauge potentials. As a by-product, free field equations for a massless gauge field are shown to possess both su ( 2 , 2 ) ∼ o ( 4 , 2 ) and sl ( 4 , R ) ∼ o ( 3 , 3 ) symmetry. The proposed formulation is well-defined in the AdS 4 background but experiences certain degeneracy in the flat limit that does not allow conformal invariant field equations for spin s > 1 gauge fields in Minkowski space. The basis model involves the doubled set of fields of all spins. It is manifestly invariant under U ( 1 ) electric–magnetic duality extended to higher spins. Reduction to a single massless field contains the equations that relate its electric and magnetic potentials which are mixed by the conformal transformations for s > 1 . We use the unfolded formulation approach recalled in the paper with some emphasis on the role of Chevalley–Eilenberg cohomology of a Lie algebra g in g-invariant field equations. This method makes it easy to guess a form of the 4 d sp ( 8 , R ) invariant massless field equations and then to extend them to the ten-dimensional sp ( 8 , R ) invariant space–time. Dynamical content of the field equations is analyzed in terms of σ − cohomology.