We argue that a spontaneous breakdown of local Weyl invariance offers a mechanism in which gravit... more We argue that a spontaneous breakdown of local Weyl invariance offers a mechanism in which gravitational interactions contribute to the generation of particle masses and their electric charge. The theory is formulated in terms of a spacetime geometry whose natural connection has both dynamic torsion and non-metricity. Its structure illuminates the role of dynamic scales used to determine measurable aspects of particle interactions and it predicts an additional neutral vector boson with electroweak properties.
A consistent variational derivation of the Majorana 4-spinor field equations coupled to Einstein'... more A consistent variational derivation of the Majorana 4-spinor field equations coupled to Einstein's theory of gravitation is given. The equivalence of the first and the second order variational field equations is explicitly demonstrated. The Lagrange multiplier 2-forms we use turn out to be precisely the Belinfante-Rosenfeld 2-forms that are needed to symmetrize the canonical energy-momentum tensor of the Majorana spinor.
An SL(2, C) gauge formulation of a scalar-tensor theory of gravitation is related to the Brans-Di... more An SL(2, C) gauge formulation of a scalar-tensor theory of gravitation is related to the Brans-Dicke theory in the absence of spinorial matter couplings although the natural connection of space-time has a torsion determined by the scalar field. Simplified scaling properties of gravitational variables are elucidated in this geometry. In the presence of matter with intrinsic spin a natural scalar-spinor coupling is exposed. A locally scale invariant limit exists without a Weyl gauge field.
We argue that by treating gravity as a local SL(2, C) gauge theory, the gauge covariant tensor fo... more We argue that by treating gravity as a local SL(2, C) gauge theory, the gauge covariant tensor formulation of field theory is entirely adequate to discuss the coupling of all gauge fields to gravity in a manner that preserves gauge covariance. In particular we make precise the notion of minimal coupling in the presence of torsion.
Newton-Cartan manifolds and the Galilei group are defined by the use of co-rank one degenerate me... more Newton-Cartan manifolds and the Galilei group are defined by the use of co-rank one degenerate metric tensor. Newton-Cartan connection is lifted to the degenerate spinor bundle over a Newton-Cartan 4manifold by the aid of degenerate spin group. Levy-Leblond equation is constructed with the lifted connection.
Impulsive, nondiverging, Petrov-Segre type-N gravitational wave solutions to a general massive th... more Impulsive, nondiverging, Petrov-Segre type-N gravitational wave solutions to a general massive three-dimensional gravity in the de Sitter, anti-de Sitter and flat Minkowski backgrounds are constructed in a unified manner by using the exterior algebra of differential forms.
We argue that the geodesic hypothesis based on auto-parallels of the Levi-Civita connection may n... more We argue that the geodesic hypothesis based on auto-parallels of the Levi-Civita connection may need refinement in theories of gravity with additional scalar fields. This argument is illustrated with a reformulation of the Brans-Dicke theory in terms of a spacetime connection with torsion determined dynamically in terms of the gradient of the Brans-Dicke scalar field. The perihelion shift in the orbit of Mercury is calculated on the alternative hypothesis that its worldline is an auto-parallel of such a connection. If scalar fields couple significantly to matter and spinless test particles move on such worldlines, current time keeping methods based on the conventional geodesic hypothesis may need refinement.
We obtain a general class of exact solutions to topologically massive gravity with or without a n... more We obtain a general class of exact solutions to topologically massive gravity with or without a negative cosmological constant. In the first case, we show that the solution is supersymmetric and asymptotically approaches the extremal BTZ black hole solution, while in the latter case it goes to flat space-time.
The question of the interpretation of Wheeler-DeWitt solutions in the context of cosmological mod... more The question of the interpretation of Wheeler-DeWitt solutions in the context of cosmological models is addressed by implementing the Hamiltonian constraint as a spinor wave equation in minisuperspace. We offer a relative probability interpretation based on a non-closed vector current in this space and a prescription for a parametrisation of classical solutions in terms of classical time. Such a prescription can accommodate classically degenerate metrics describing manifolds with signature change. The relative probability density, defined in terms of a Killing vector of the Dewitt metric on minisuperspace, should permit one to identify classical loci corresponding to geometries for a classical manifold. This interpretation is illustrated in the context of a quantum cosmology model for two-dimensional dilaton gravity.
We investigate complex quaternion-valued exterior differential forms over 4-dimensional Lorentzia... more We investigate complex quaternion-valued exterior differential forms over 4-dimensional Lorentzian spacetimes and explore Weyl spinor fields as minimal left ideals within the complex quaternion algebra. The variational derivation of the coupled Einstein-Weyl equations from an action is presented, and the resulting field equations for both first and second order variations are derived and simplified. Exact plane symmetric solutions of the Einstein-Weyl equations are discussed, and two families of exact solutions describing left-moving and right-moving neutrino plane waves are provided. The study highlights the significance of adjusting a quartic selfcoupling of the Weyl spinor in the action to ensure the equivalence of the field equations.
A non-minimal coupling of Weyl curvatures to electromagnetic fields is considered in Brans-Dicke-... more A non-minimal coupling of Weyl curvatures to electromagnetic fields is considered in Brans-Dicke-Maxwell theory. The gravitational field equations are formulated in a Riemannian spacetime where the spacetime torsion is constrained to zero by the method of Lagrange multipliers in the language of exterior differential forms. The significance and ramifications of non-minimal couplings to gravity are examined in a pp-wave spacetime.
Bertotti-Robinson spacetimes are topologically AdS2 × S 2 and described by a conformally flat met... more Bertotti-Robinson spacetimes are topologically AdS2 × S 2 and described by a conformally flat metric. Together with the Coulomb electric potential, they provide a class of static, geodetically complete Einstein-Maxwell solutions. We show here that the Bertotti-Robinson metric together with Wu-Yang magnetic pole potentials give a class of static solutions of a system of non-minimally coupled Einstein-Yang-Mills equations that may be relevant for investigating vacuum polarization effects in a first order perturbative approach to quantum fields.
We linearize vacuum Einstein field equations with a cosmological constant around a curved backgro... more We linearize vacuum Einstein field equations with a cosmological constant around a curved background to elaborate on the reconstruction of the Abbott-Deser charges and incorporate a spin connection into the definition using the algebra of differential forms on a given curved background spacetime.
The gravitational field equations of Brans-Dicke theory are given in a 4-dimensional non-Riemanni... more The gravitational field equations of Brans-Dicke theory are given in a 4-dimensional non-Riemannian space-time with torsion in the language of exterior differential forms. A class of pp-wave metrics together with the Brans-Dicke scalar field are used to derive the autoparallel equations of motion for non-spinning test masses. These are compared with the geodesic equations of motion and the differences are pointed out. The effects of the gradient of the Brans-Dicke scalar on the geodesic deviation equations in this non-Riemannian setting are also discussed.
The low-energy (bosonic "heterotic") string theory is interpreted as a universal limit of the Kal... more The low-energy (bosonic "heterotic") string theory is interpreted as a universal limit of the Kaluza-Klein reduction when the dimension of an internal space goes to infinity. We show that such an approach is helpful in obtaining classical solutions of the string model. As a particular application, we obtain new exact static solutions for the two-dimensional effective string model.
We point our a new class of solutions of the supersymmetrlc Yang-Mdls equations. This class provi... more We point our a new class of solutions of the supersymmetrlc Yang-Mdls equations. This class provides solutions which cannot be generated from the solutions of the ordinary Yang-Mdls equations by flmte supersymmetry transformations and contains the supersymmetric generahzatlon of the non-abehan plane waves.
Sandwich gravitational waves are given globally in terms of step functions at the boundaries. Lin... more Sandwich gravitational waves are given globally in terms of step functions at the boundaries. Linearized Einstein–Weyl equations are solved exactly in this background in Rosen coordinates. Depending on the geometry and composition of the sandwich wave, the neutrino’s energy-momentum redistributes itself. At the test field level, since the background will not change, the neutrino’s energy density in particular will show variations between positive and negative extrema when crossing the sandwich wave. This may reveal facts about the weakly interacting neutrinos in cosmology.
A consistent variational derivation of the Majorana 4-spinor field equations coupled to Einstein'... more A consistent variational derivation of the Majorana 4-spinor field equations coupled to Einstein's theory of gravitation is given. The equivalence of the first and the second order variational field equations is explicitly demonstrated. The Lagrange multiplier 2-forms we use turn out to be precisely the Belinfante-Rosenfeld 2-forms that are needed to symmetrize the canonical energy-momentum tensor of the Majorana spinor.
Strongly self-dual Yang-Mills fields in even dimensional spaces are characterised by a set of con... more Strongly self-dual Yang-Mills fields in even dimensional spaces are characterised by a set of constraints on the eigenvalues of the Yang-Mills fields F µν. We derive a topological bound on R 8 , M (F, F) 2 ≥ k M p 2
International Journal of Geometric Methods in Modern Physics, Jul 24, 2023
In the conventional formulation of general relativity, gravity is represented by the metric curva... more In the conventional formulation of general relativity, gravity is represented by the metric curvature of Riemannian geometry. There are also alternative formulations in flat affine geometries, wherein the gravitational dynamics is instead described by torsion and nonmetricity. These so called general teleparallel geometries may also have applications in material physics, such as the study of crystal defects. In this work, we explore the general teleparallel geometry in the language of differential forms. We discuss the special cases of metric and symmetric teleparallelisms, clarify the relations between formulations with different gauge fixings and without gauge fixing, and develop a method of recasting Riemannian into teleparallel geometries. As illustrations of the method, exact solutions are presented for the generic quadratic theory in 2, 3 and 4 dimensions.
We argue that a spontaneous breakdown of local Weyl invariance offers a mechanism in which gravit... more We argue that a spontaneous breakdown of local Weyl invariance offers a mechanism in which gravitational interactions contribute to the generation of particle masses and their electric charge. The theory is formulated in terms of a spacetime geometry whose natural connection has both dynamic torsion and non-metricity. Its structure illuminates the role of dynamic scales used to determine measurable aspects of particle interactions and it predicts an additional neutral vector boson with electroweak properties.
A consistent variational derivation of the Majorana 4-spinor field equations coupled to Einstein'... more A consistent variational derivation of the Majorana 4-spinor field equations coupled to Einstein's theory of gravitation is given. The equivalence of the first and the second order variational field equations is explicitly demonstrated. The Lagrange multiplier 2-forms we use turn out to be precisely the Belinfante-Rosenfeld 2-forms that are needed to symmetrize the canonical energy-momentum tensor of the Majorana spinor.
An SL(2, C) gauge formulation of a scalar-tensor theory of gravitation is related to the Brans-Di... more An SL(2, C) gauge formulation of a scalar-tensor theory of gravitation is related to the Brans-Dicke theory in the absence of spinorial matter couplings although the natural connection of space-time has a torsion determined by the scalar field. Simplified scaling properties of gravitational variables are elucidated in this geometry. In the presence of matter with intrinsic spin a natural scalar-spinor coupling is exposed. A locally scale invariant limit exists without a Weyl gauge field.
We argue that by treating gravity as a local SL(2, C) gauge theory, the gauge covariant tensor fo... more We argue that by treating gravity as a local SL(2, C) gauge theory, the gauge covariant tensor formulation of field theory is entirely adequate to discuss the coupling of all gauge fields to gravity in a manner that preserves gauge covariance. In particular we make precise the notion of minimal coupling in the presence of torsion.
Newton-Cartan manifolds and the Galilei group are defined by the use of co-rank one degenerate me... more Newton-Cartan manifolds and the Galilei group are defined by the use of co-rank one degenerate metric tensor. Newton-Cartan connection is lifted to the degenerate spinor bundle over a Newton-Cartan 4manifold by the aid of degenerate spin group. Levy-Leblond equation is constructed with the lifted connection.
Impulsive, nondiverging, Petrov-Segre type-N gravitational wave solutions to a general massive th... more Impulsive, nondiverging, Petrov-Segre type-N gravitational wave solutions to a general massive three-dimensional gravity in the de Sitter, anti-de Sitter and flat Minkowski backgrounds are constructed in a unified manner by using the exterior algebra of differential forms.
We argue that the geodesic hypothesis based on auto-parallels of the Levi-Civita connection may n... more We argue that the geodesic hypothesis based on auto-parallels of the Levi-Civita connection may need refinement in theories of gravity with additional scalar fields. This argument is illustrated with a reformulation of the Brans-Dicke theory in terms of a spacetime connection with torsion determined dynamically in terms of the gradient of the Brans-Dicke scalar field. The perihelion shift in the orbit of Mercury is calculated on the alternative hypothesis that its worldline is an auto-parallel of such a connection. If scalar fields couple significantly to matter and spinless test particles move on such worldlines, current time keeping methods based on the conventional geodesic hypothesis may need refinement.
We obtain a general class of exact solutions to topologically massive gravity with or without a n... more We obtain a general class of exact solutions to topologically massive gravity with or without a negative cosmological constant. In the first case, we show that the solution is supersymmetric and asymptotically approaches the extremal BTZ black hole solution, while in the latter case it goes to flat space-time.
The question of the interpretation of Wheeler-DeWitt solutions in the context of cosmological mod... more The question of the interpretation of Wheeler-DeWitt solutions in the context of cosmological models is addressed by implementing the Hamiltonian constraint as a spinor wave equation in minisuperspace. We offer a relative probability interpretation based on a non-closed vector current in this space and a prescription for a parametrisation of classical solutions in terms of classical time. Such a prescription can accommodate classically degenerate metrics describing manifolds with signature change. The relative probability density, defined in terms of a Killing vector of the Dewitt metric on minisuperspace, should permit one to identify classical loci corresponding to geometries for a classical manifold. This interpretation is illustrated in the context of a quantum cosmology model for two-dimensional dilaton gravity.
We investigate complex quaternion-valued exterior differential forms over 4-dimensional Lorentzia... more We investigate complex quaternion-valued exterior differential forms over 4-dimensional Lorentzian spacetimes and explore Weyl spinor fields as minimal left ideals within the complex quaternion algebra. The variational derivation of the coupled Einstein-Weyl equations from an action is presented, and the resulting field equations for both first and second order variations are derived and simplified. Exact plane symmetric solutions of the Einstein-Weyl equations are discussed, and two families of exact solutions describing left-moving and right-moving neutrino plane waves are provided. The study highlights the significance of adjusting a quartic selfcoupling of the Weyl spinor in the action to ensure the equivalence of the field equations.
A non-minimal coupling of Weyl curvatures to electromagnetic fields is considered in Brans-Dicke-... more A non-minimal coupling of Weyl curvatures to electromagnetic fields is considered in Brans-Dicke-Maxwell theory. The gravitational field equations are formulated in a Riemannian spacetime where the spacetime torsion is constrained to zero by the method of Lagrange multipliers in the language of exterior differential forms. The significance and ramifications of non-minimal couplings to gravity are examined in a pp-wave spacetime.
Bertotti-Robinson spacetimes are topologically AdS2 × S 2 and described by a conformally flat met... more Bertotti-Robinson spacetimes are topologically AdS2 × S 2 and described by a conformally flat metric. Together with the Coulomb electric potential, they provide a class of static, geodetically complete Einstein-Maxwell solutions. We show here that the Bertotti-Robinson metric together with Wu-Yang magnetic pole potentials give a class of static solutions of a system of non-minimally coupled Einstein-Yang-Mills equations that may be relevant for investigating vacuum polarization effects in a first order perturbative approach to quantum fields.
We linearize vacuum Einstein field equations with a cosmological constant around a curved backgro... more We linearize vacuum Einstein field equations with a cosmological constant around a curved background to elaborate on the reconstruction of the Abbott-Deser charges and incorporate a spin connection into the definition using the algebra of differential forms on a given curved background spacetime.
The gravitational field equations of Brans-Dicke theory are given in a 4-dimensional non-Riemanni... more The gravitational field equations of Brans-Dicke theory are given in a 4-dimensional non-Riemannian space-time with torsion in the language of exterior differential forms. A class of pp-wave metrics together with the Brans-Dicke scalar field are used to derive the autoparallel equations of motion for non-spinning test masses. These are compared with the geodesic equations of motion and the differences are pointed out. The effects of the gradient of the Brans-Dicke scalar on the geodesic deviation equations in this non-Riemannian setting are also discussed.
The low-energy (bosonic "heterotic") string theory is interpreted as a universal limit of the Kal... more The low-energy (bosonic "heterotic") string theory is interpreted as a universal limit of the Kaluza-Klein reduction when the dimension of an internal space goes to infinity. We show that such an approach is helpful in obtaining classical solutions of the string model. As a particular application, we obtain new exact static solutions for the two-dimensional effective string model.
We point our a new class of solutions of the supersymmetrlc Yang-Mdls equations. This class provi... more We point our a new class of solutions of the supersymmetrlc Yang-Mdls equations. This class provides solutions which cannot be generated from the solutions of the ordinary Yang-Mdls equations by flmte supersymmetry transformations and contains the supersymmetric generahzatlon of the non-abehan plane waves.
Sandwich gravitational waves are given globally in terms of step functions at the boundaries. Lin... more Sandwich gravitational waves are given globally in terms of step functions at the boundaries. Linearized Einstein–Weyl equations are solved exactly in this background in Rosen coordinates. Depending on the geometry and composition of the sandwich wave, the neutrino’s energy-momentum redistributes itself. At the test field level, since the background will not change, the neutrino’s energy density in particular will show variations between positive and negative extrema when crossing the sandwich wave. This may reveal facts about the weakly interacting neutrinos in cosmology.
A consistent variational derivation of the Majorana 4-spinor field equations coupled to Einstein'... more A consistent variational derivation of the Majorana 4-spinor field equations coupled to Einstein's theory of gravitation is given. The equivalence of the first and the second order variational field equations is explicitly demonstrated. The Lagrange multiplier 2-forms we use turn out to be precisely the Belinfante-Rosenfeld 2-forms that are needed to symmetrize the canonical energy-momentum tensor of the Majorana spinor.
Strongly self-dual Yang-Mills fields in even dimensional spaces are characterised by a set of con... more Strongly self-dual Yang-Mills fields in even dimensional spaces are characterised by a set of constraints on the eigenvalues of the Yang-Mills fields F µν. We derive a topological bound on R 8 , M (F, F) 2 ≥ k M p 2
International Journal of Geometric Methods in Modern Physics, Jul 24, 2023
In the conventional formulation of general relativity, gravity is represented by the metric curva... more In the conventional formulation of general relativity, gravity is represented by the metric curvature of Riemannian geometry. There are also alternative formulations in flat affine geometries, wherein the gravitational dynamics is instead described by torsion and nonmetricity. These so called general teleparallel geometries may also have applications in material physics, such as the study of crystal defects. In this work, we explore the general teleparallel geometry in the language of differential forms. We discuss the special cases of metric and symmetric teleparallelisms, clarify the relations between formulations with different gauge fixings and without gauge fixing, and develop a method of recasting Riemannian into teleparallel geometries. As illustrations of the method, exact solutions are presented for the generic quadratic theory in 2, 3 and 4 dimensions.
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Papers by Tekin Dereli