On Gauge Invariant Theories of Gravitation

Theories of gravitation are called gauge invariant if the invariance of the gravitational field lagrangian with respect to gauge transformations of the gravitational field variables is independend of the invariance of this lagrangian with respect to the Einstein group of general coordinate transformations. They are bimetric theories because the coordinate covariance is ensured by constructing scalar densities relative to a globally flat background metric. Such a theory is represented by the PAUL-FIERZ equations for massless spin 2 particles. But this theory is inconsistent if nongravitational matter is enclosed as a source. All attempts to overcome this inconsistancy preserving gauge invariance lead to Einstein's GRT. We review this problem and compare the situation with a theory proposed by LOGUNOV showing that he overcomes the inconsistency of linear Einstein's equations by replacing the field variables by a gauge invariant combination of new ones, which turns out to be the first order form of v. FREUD'S superpotential.     

Metric-affine variational principles in general relativity II. Relaxation of the Riemannian constraint

We continue our investigation of a variational principle for general relativity in which the metric tensor and the (asymmetric) linear connection are varied independently. As in Part I, the matter Lagrangian is minimally coupled to the connection and the gravitational Lagrangian is taken to be the curvature scalar, but we now relax the Riemannian constraint as far as possible—that is, as far as the projective invariance of the assumed gravitational Lagrangian will allow. The outcome of this procedure is a gravitational theory formulated in a volume-preserving space-time (i.e., with torsion and tracefree nonmetricity). The vanishing of the trace of the nonmetricity is due to the remaining vector constraint. We also discuss the physical significance of the relaxation of the Riemannian constraint, the possible relaxation of the vector constraint, the notion of the hypermomentum current, and its possible relation to elementary particle physics.     

Slightly Bimetric Gravitation

The inclusion of a flat metric tensor in gravitation permits the formulation of a gravitational stress-energy tensor and the formal derivation of general relativity from a linear theory in flat spacetime. Building on the works of Kraichnan and Deser, we present such a derivation using universal coupling and gauge invariance.Next we slightly weaken the assumptions of universal coupling and gauge invariance, obtaining a larger "slightly bimetric" class of theories, in which the Euler-Lagrange equations depend only on a curved metric, matter fields, and the determinant of the flat metric. The theories are equivalent to generally covariant theories with an arbitrary cosmological constant and an arbitrarily coupled scalar field, which can serve as an inflaton or dark matter.The question of the consistency of the null cone structures of the two metrics is addressed.     

Gravitational Gauge Interactions of Scalar Field

Quantum gauge theory of gravity is formulated based on gauge principle. Because the Lagrangian has strict local gravitational gauge symmetry, gravitational gauge theory is a perturbatively renormalizable quantum theory. Gravitational gauge interactions of scalar field are studied in this paper. In quantum gauge theory of gravity, scalar field minimal couples to gravitational field through gravitational gauge covariant derivative. Comparing the Lagrangian for scalar field in quantum gauge theory of gravity with the corresponding Lagrangian in quantum fields in curved space-time, the definition for metric in curved space-time in geometry picture of gravity can be obtained, which is expressed by gravitational gauge field. In classical level, the Lagrangian and Hamiltonian approaches are also discussed.     

Two kinds of rotation: An argument for torsion

There are now many theories of gravity with a torsion field as well as the usual metric field. One of the arguments for allowing torsion is based upon a gauge theory analogy. The purpose of this paper is to clarify exactly which symmetries are being gauged in this process. The principal observation is that special relativity is invariant under two different kinds of Lorentz transformations. The first type rotate the fields and move them from one point to another in space-time. The second type merely rotate the fields at each point without changing their location. To gauge both types of rotations requires a torsion field as well as a metric field.This essay received honorable mention from the Gravity Research Foundation for the year 1980 (Ed.).     

Gravitational Gauge Interactions of Scalar Field

Quantum gauge theory of gravity is formulated based on gauge principle. Because the Lagrangian hasstrict local gravitational gauge symmetry, gravitational gauge theory is a perturbatively renormalizable quantum theory.Gravitational gauge interactions of scalar field are studied in this paper. In quantum gauge theory of gravity, scalar fieldminimal couples to gravitational field through gravitational gauge covariant derivative. Comparing the Lagrangian forscalar field in quantum gauge theory of gravity with the corresponding Lagrangian in quantum fields in curved space-time, the definition for metric in curved space-time in geometry picture of gravity can be obtained, which is expressedby gravitational gauge field. In classical level, the Lagrangian and Hamiltonian approaches are also discussed.     

Some remarks on zeta function regularization in curved space-times

The question of to what extent zeta function regularization respects the invariances of a quantum field theory in a background gravitational field is investigated. It is shown that zeta function regularization provides a generalization to curved space-time of analytic propagator regularization which is known not to respect gauge invariance. Furthermore, a study of the regularized stress tensor of a conformally invariant scalar field indicates that both conformai and general coordinate invariance are violated.     

Local Scale Invariance and General Relativity

According to the theory of unimodular relativity developed by Anderson and Finkelstein, the equations of general relativity with a cosmological constant are composed of two independent equations, one which determines the null-cone structure of space-time, another which determines the measure structure of space-time. The field equations that follow from the restricted variational principle of this version of general relativity only determine the null-cone structure and are globally scale-invariant and scale-free. We show that the electromagnetic field may be viewed as a compensating gauge field that guarantees local scale invariance of these field equations. In this way, Weyl's geometry is revived. However, the two principle objections to Weyl's theory do not apply to the present formulation: the Lagrangian remains first order in the curvature scalar and the nonintegrability of length only applies to the null-cone structure.     

Conformally Flat Spacetimes and Weyl Frames

We discuss the concepts of Weyl and Riemann frames in the context of metric theories of gravity and state the fact that they are completely equivalent as far as geodesic motion is concerned. We apply this result to conformally flat spacetimes and show that a new picture arises when a Riemannian spacetime is taken by means of geometrical gauge transformations into a Minkowskian flat spacetime. We find out that in the Weyl frame gravity is described by a scalar field. We give some examples of how conformally flat spacetime configurations look when viewed from the standpoint of a Weyl frame. We show that in the non-relativistic and weak field regime the Weyl scalar field may be identified with the Newtonian gravitational potential. We suggest an equation for the scalar field by varying the Einstein-Hilbert action restricted to the class of conformally-flat spacetimes. We revisit Einstein and Fokker’s interpretation of Nordstr?m scalar gravity theory and draw an analogy between this approach and the Weyl gauge formalism. We briefly take a look at two-dimensional gravity as viewed in the Weyl frame and address the question of quantizing a conformally flat spacetime by going to the Weyl frame.     

General relativity as a conformally invariant scalar gauge field theory

The global symmetry implied by the fact that one can multiply all masses with a common constant is made into a local, gauge symmetry. The matter action then becomes Conformally invariant and it seems natural to choose for the corresponding scalar gauge field the action for a conformally invariant (massless) scalar field. The resulting conformally invariant theory turns out to be equivalent to general relativity. Since this means that the usual Einstein-Hilbert action is not, in fact, a true gauge action for the space-time geometry, the full theory ought to be supplied with such a term. Gauge-theoretic arguments and conformal invariance requirements dictate its form.     

Bianchi identities and the automatic conservation of energy-momentum and angular momentum in general-relativistic field theories

Automatic conservation of energy-momentum and angular momentum is guaranteed in a gravitational theory if, via the field equations, the conservation laws for the material currents are reduced to the contracted Bianchi identities. We first execute an irreducible decomposition of the Bianchi identities in a Riemann-Cartan space-time. Then, starting from a Riemannian space-time with or without torsion, we determine those gravitational theories which have automatic conservation: general relativity and the Einstein-Cartan-Sciama-Kibble theory, both with cosmological constant, and the nonviable pseudoscalar model. The Poincaré gauge theory of gravity, like gauge theories of internal groups, has no automatic conservation in the sense defined above. This does not lead to any difficulties in principle. Analogies to 3-dimensional continuum mechanics are stressed throughout the article.     

On the Interaction of Fermion and Boson Fields with the Gravitational Field

We point out that the gravitational field taken by itself cannot be considered as a gauge field. Only an affinity and not a metric can serve as a gauge field. Originally, metric and affinity are completely independent of each other. This fact allows in a natural way to formulate a restricted principle of relativity, according to which only fermion fields may show that there exist a priori distinguished frames of reference. Furthermore, we can couple the gravitational field to boson and fermion fields such that the flat metric or tetrads orthonormalized with respect to this flat metric appearing in the special relativistic matter Lagrangian, are replaced by a Riemannian metric and tetrads orthonormalized with respect to this metric (principle of most minimal gravitational coupling). This coupling principle is a strong restriction on the existence of independent boson fields. Only scalar and vector fields and their different pseudoquantities are possible as independent fields. Boson fields of higher rank are to be considered as fusions of these (pseudo)scalar and (pseudo)vector fields. Theire field equations follow from those of the (pseudo)scalar and (pseudo)vector fields.     

Gauge Gravitational Field in a Fractal Space-Time

Considering the fractal structure of space-time, the scale relativity theory in the topological dimension D_T=2 is built. In such a conjecture, the geodesics of this space-time imply the hydrodynamic model of the quantum mechanics. Subsequently, the gauge gravitational field on a fractal space-time is given. Then, the gauge group, the gauge-covariant derivative, the strength tensor of the gauge field, the gauge-invariant Lagrangean, the field equations of the gauge potentials and the gauge energy-momentum tensor are determined. Finally, using this model, a Reissner-Nordström type metric is obtained.     

The Brans-Dicke scalar field in Einstein-Cartan theory

In the framework of Einstein-Cartan (EC) theory, the Brans-Dicke (BD) theory is considered and it is found that a scalar field nonminimally coupled to the gravitational field gives rise to torsion, even though the scalar field has zero spin. The metric equations stay the same if the coupling constant is rescaled, but the equations of motion of a test particle, derived from the conservation equations, differ from those of the usual BD theory without torsion. The gravitational red-shift value differs considerably from the usual prediction of general theory of relativity (GTR), and rules out the possibility of a torsion version of BD theory for<6.     

Quadratic Poincaré gauge theory of gravity: A comparison with the general relativity theory

The classical dynamics of the gravitational field in the Poincaré gauge theory is studied. The most general Lagrangian quadratic in curvature and torsion is considered. The relevant field equations and their solutions are analyzed in detail, with particular emphasis on the comparison of the Poincaré gauge models with the general relativity theory. We investigate correspondence between the spaces of exact solutions of these theories, both in the presence and absence of material sources, and with or without torsion. Some new exact solutions are obtained without the use of the double duality ansatz. The weak-field approximation is discussed, and gravitational radiation is considered.     

Novel ansatzes and scalar quantities in gravito-electromagnetism

In this work, we focus on the theory of gravito-electromagnetism (GEM)—the theory that describes the dynamics of the gravitational field in terms of quantities met in electromagnetism—and we propose two novel forms of metric perturbations. The first one is a generalisation of the traditional GEM ansatz, and succeeds in reproducing the whole set of Maxwell’s equations even for a dynamical vector potential \(\mathbf {A}\). The second form, the so-called alternative ansatz, goes beyond that leading to an expression for the Lorentz force that matches the one of electromagnetism and is free of additional terms even for a dynamical scalar potential \(\varPhi \). In the context of the linearised theory, we then search for scalar invariant quantities in analogy to electromagnetism. We define three novel, 3rd-rank gravitational tensors, and demonstrate that the last two can be employed to construct scalar quantities that succeed in giving results very similar to those found in electromagnetism. Finally, the gauge invariance of the linearised gravitational theory is studied, and shown to lead to the gauge invariance of the GEM fields \(\mathbf {E}\) and \(\mathbf {B}\) for a general configuration of the arbitrary vector involved in the coordinate transformations.     

On a new variational principle in general relativity and the energy of the gravitational field

A new variational principle based on the affine connection in space-time is proposed. This leads to a new formulation of general relativity. The gravitational field is a field of inertial frames in space-time. The metricg appears as a momentum canonically conjugate to the gravitational field. In the case of simple matter fields, e.g., scalar fields, electromagnetic fields, Proca fields, or hydrodynamical matter, the new formulation is equivalent to the traditional one. A new formulation of conservation laws is proposed.     

On Gravitational Energy in Newtonian Theories

There are well-known problems associated with the idea of (local) gravitational energy in general relativity. We offer a new perspective on those problems by comparison with Newtonian gravitation, and particularly geometrized Newtonian gravitation (i.e., Newton–Cartan theory). We show that there is a natural candidate for the energy density of a Newtonian gravitational field. But we observe that this quantity is gauge dependent, and that it cannot be defined in the geometrized (gauge-free) theory without introducing further structure. We then address a potential response by showing that there is an analogue to the Weyl tensor in geometrized Newtonian gravitation.