Sparling Two-Forms, the Conformal Factor and the Gravitational Energy Density of the Teleparallel Equivalent of General Relativity

It has been shown recently that within the framework of the teleparallel equivalent of general relativity (TEGR) it is possible to define the energy density of the gravitational field in a unique way. The tegr amounts to an alternative formulation of Einstein's general relativity, not to an alternative gravity theory. The localizability of the gravitational energy has been investigated in a number of spacetimes with distinct topologies, and the outcome of these analyses agree with previously known results regarding the exact expression of the gravitational energy, and/or with the specific properties of the spacetime manifold. In this article we establish a relationship between the expression of the gravitational energy density of the TEGR and the Sparling two-forms, which are known to be closely connected with the gravitational energy. We will also show that our expression of energy yields the correct value of gravitational mass contained in the conformal factor of the metric field.     

Spherical gravitational collapse with photon emission and a generalized Schwarzchild interior solution

The general dynamical equations for perfect fluid filled spheres with an outward flux of photons are derived. The vital role played by the energy density of the free gravitational field in accelerating photon production has been emphasized. It is pointed out that even when the material energy density is finite, the energy density of the free gravitational field can take infinitely large values resulting in vanishing surface area of the star. A generalized Schwarzschild interior solution with conformally flat geometry but with photon emission has been obtained. It is pointed out that the interior conformal coordinate system bears a strong resemblance to the exterior Krushkal coordinates. It is shown that for spherical star the invariant velocity of the fluid particles, falling towards the centre, is proportional to its radius suggesting that the outer envelopes collapse at a faster rate than the core part. It is shown that the interior radiating solution can be matched with generalized Schwarzchild exterior solution.     

Spherically symmetric free fall collapse

The general dynamical equations for spherical gravitational collapse are derived by introducing the eigenvalue of the conformal Weyl tensor in the 2-2 component of the Einstein tensor and assuming the material content of the models to be a perfect fluid. Since this eigenvalue is coupled always with the material energy density, it has been interpreted as theenergy density of the free gravitational field whose presence is related with anisotropy and inhomogeneity. As a particular case, the collapse of a spherically symmetric dust (zero pressure) with vanishing radial acceleration (free fall collapse) is discussed. It is shown that the model is inhomogeneous with non-vanishing shear of the congruence of world lines of the dust particles. The model contains gravitational radiation by Szekere’s criterion since both shear invariant and the spatial gradient of density are non-vanishing. This is in contrast to the Oppenheimer-Synder model for which both the above mentioned characteristics are absent. A particular solution which is anisotropic and inhomogeneous has been given to prove the emission of gravitational radiation by the freely falling dust and in this case the energy density of the free gravitational field contains a typeN term superposed on the coulombian field.     

Conformally equivalent metrics in bimetric general relativity

Space-times conformal to physical space-time are considered, assuming the nongravitational energy is conservative after the conformal transformation. Necessary and sufficient conditions satisfied by the conformal factor axe found for a given type of transformation of the energy tensor. The weak gravitational field is defined and the coordinate conditions for the existence of conformal factors in such a field are obtained.     

On the conformal transformations of metrics in Rosen's bimetric gravitation theory

The conformal transformations of the background metric are considered and the transformed expressions of the gravitational energy and of the gravitational field equations are obtained. The conditions of conservation of the nongravitational energy are formulated with respect to the transformed metric. Then the background and the physical metrics are both subjected to transformations and the relationship between the corresponding conformal factors analyzed, particularly in the case of conserved nongravitational energy.     

Some properties of the gravitational fields of a hydrodynamical fluid and dust

Conditions are obtained on the four velocity of the particles of a medium in the gravitational fields of a hydrodynamical fluid and dust, and the nature of the motion of the medium in these fields is investigated. It is shown that the gravitational fields of a hydrodynamical fluid and dust do not admit conformal mapping onto Einstein spaces which are not conformally flat spaces (spaces of constant curvature).     

Comprehensive solution to the cosmological constant,zero-point energy,and quantum gravity problems

We present a solution to the cosmological constant, the zero-point energy, and the quantum gravity problems within a single comprehensive framework. We show that in quantum theories of gravity in which the zero-point energy density of the gravitational field is well-defined, the cosmological constant and zero-point energy problems solve each other by mutual cancellation between the cosmological constant and the matter and gravitational field zero-point energy densities. Because of this cancellation, regulation of the matter field zero-point energy density is not needed, and thus does not cause any trace anomaly to arise. We exhibit our results in two theories of gravity that are well-defined quantum-mechanically. Both of these theories are locally conformal invariant, quantum Einstein gravity in two dimensions and Weyl-tensor-based quantum conformal gravity in four dimensions (a fourth-order derivative quantum theory of the type that Bender and Mannheim have recently shown to be ghost-free and unitary). Central to our approach is the requirement that any and all departures of the geometry from Minkowski are to be brought about by quantum mechanics alone. Consequently, there have to be no fundamental classical fields, and all mass scales have to be generated by dynamical condensates. In such a situation the trace of the matter field energy-momentum tensor is zero, a constraint that obliges its cosmological constant and zero-point contributions to cancel each other identically, no matter how large they might be. In our approach quantization of the gravitational field is caused by its coupling to quantized matter fields, with the gravitational field not needing any independent quantization of its own. With there being no a priori classical curvature, one does not have to make it compatible with quantization.     

Gravitational coupling and dynamical reduction of the cosmological constant

We introduce a dynamical model to reduce a large cosmological constant to a sufficiently small value. The basic ingredient in this model is a distinction which has been made between the two unit systems used in cosmology and particle physics. We have used a conformal invariant gravitational model to define a particular conformal frame in terms of large scale properties of the universe. It is then argued that the contributions of mass scales in particle physics to the vacuum energy density should be considered in a different conformal frame. In this manner, a decaying mechanism is presented in which the conformal factor appears as a dynamical field and plays a key role to relax a large effective cosmological constant. Moreover, we argue that this model also provides a possible explanation for the coincidence problem.     

Dynamical Conformal Transformation and Classical Euclidean Wormholes

We investigate the necessary condition for the existence of classical Euclidean wormholes in a conformally non-invariant gravitational model minimally coupled to an scalar field. It is shown that while the original Ricci tensor with positive eigenvalues does not allow the Euclidean wormholes to occur, under dynamical conformal transformations the Ricci tensor, with respect to the original metric, is dynamically coupled with the conformal field and its eigenvalues may become negative allowing the Euclidean wormholes to occur. Therefore, it is conjectured that dynamical conformal transformations may provide us with effective forms of matter sources leading to Euclidean wormholes in conformally non-invariant systems.     

Wormhole supported by dark energy admitting conformal motion

In this article, we study the possibility of sustaining static and spherically symmetric traversable wormhole geometries admitting conformal motion in Einstein gravity, which presents a more systematic approach to search a relation between matter and geometry. In wormhole physics, the presence of exotic matter is a fundamental ingredient and we show that this exotic source can be dark energy type which support the existence of wormhole spacetimes. In this work we model a wormhole supported by dark energy which admits conformal motion. We also discuss the possibility of the detection of wormholes in the outer regions of galactic halos by means of gravitational lensing. Studies of the total gravitational energy for the exotic matter inside a static wormhole configuration are also performed.     

Variational principle for gravitational equations of the Bianchi identity type

It is shown that a square invariant of the Weyl conformal curvature tensor can lead to a Lagrangian in a variational principle for a gravitational equation in vacuum of the Bianchi identity type which is compatible with the Einstein equation. Moreover we show that such a Lagrangian implicitly includes a conformally invariant theory characterized by two gauge fields and the metric tensor.     

Gravitational field of global monopole with semi-classical effects

We find an exact solution for the space-time of a global monopole by using the vacuum expectation value of the stress energy tensor due to an arbitrary collection of conformal mass less free quantum fields as a source. In a particular situation, the solution is shown to possess an interesting feature like ‘wormholes’ space-time. The monopole exerts no gravitational force on the surrounding matter.     

Ratios of magnetic charge to charge and of magnetic mass to mass

It is shown that, unlike the case of (vacuum) solutions describing isolated bodies, conformal Killing fields are not excluded by the structure of vacuum gravitational magnetic monopoles at null infinity. The resulting dilation must be constant. This brings support to the viewpoint that such solutions might have a role to play in the understanding of gravitational entropy and time's arrow. If, in addition, a Maxwellian magnetic monopole (Dirac string singularity) is available, the ratio of the total magnetic charge (magnetic mass) over the total electric charge (mass) can be identified. This common feature between the gravitational and the electromagnetic interaction finds its origin in the space-time topology.     

A unified conformal model for fundamental interactions without dynamical Higgs field

A Higgsless model for strong, electroweak and gravitational interactions is proposed. This model is based on the local symmetry group SU(3)×SU(2)_L×U(1)×C,where C is the local conformal symmetry group. The natural minimal conformally invariant form of total Lagrangian is postulated. It contains all standard model fields and gravitational interaction. Using the unitary gauge and the conformal scale fixing conditions, we can eliminate all four real components of the Higgs doublet in this model. However, the masses of vector mesons, leptons, and quarks are automatically generated and are given by the same formulas as in the conventional standard model. In this manner one gets the mass generation without the mechanism of spontaneous symmetry breaking and without the remaining real dynamical Higgs field. The gravitational sector is analyzed, and it is shown that the model admits in the classical limit the Einsteinian form of gravitational interactions.     

Vertex Operators in 4D Quantum Gravity Formulated as CFT

We study vertex operators in 4D conformal field theory derived from quantized gravity, whose dynamics is governed by the Wess-Zumino action by Riegert and the Weyl action. Conformal symmetry is equal to diffeomorphism symmetry in the ultraviolet limit, which mixes positive-metric and negative-metric modes of the gravitational field and thus these modes cannot be treated separately in physical operators. In this paper, we construct gravitational vertex operators such as the Ricci scalar, defined as space-time volume integrals of them are invariant under conformal transformations. Short distance singularities of these operator products are computed and it is shown that their coefficients have physically correct signs. Furthermore, we show that conformal algebra holds even in the system perturbed by the cosmological constant vertex operator as in the case of the Liouville theory shown by Curtright and Thorn.     

Non-local Effects of Conformal Anomaly

It is shown that the nonlocal anomalous effective actions corresponding to the quantum breaking of the conformal symmetry can lead to observable modifications of Einstein’s equations. The fact that Einstein’s general relativity is in perfect agreement with all observations including cosmological or recently observed gravitational waves imposes strong restrictions on the field content of possible extensions of Einstein’s theory: all viable theories should have vanishing conformal anomalies. It is shown that a complete cancellation of conformal anomalies in \(D=4\) for both the \(C^2\) invariant and the Euler (Gauss–Bonnet) invariant can only be achieved for N-extended supergravity multiplets with \(N \ge 5\).     

Gravitational anomalies

The effective action for fermions moving in external gravitational and gauge fields is analyzed in terms of the corresponding external field propagator. The central object in our approach is the covariant energy-momentum tensor which is extracted from the regular part of the propagator at short distances. It is shown that the Lorentz anomaly, the conformal anomaly and the gauge anomaly can be expressed in terms of the local polynomials which determine the singular part of the propagator. (There are no coordinate anomalies.) Except for the conformal anomaly, for which we give explicit representations only ind<=4, we consider an arbitrary number of dimensions.     

Potential energy in quantum gravity

We give a general expression for the static potential energy of the gravitational interaction of two massive particles, in terms of an invariant vacuum expectation value of the quantized gravitational field. This formula holds for functional integral formulations of Euclidean quantum gravity, regularized to avoid the conformal instability. It could be regarded as the analogue of the Wilson loop of gauge theories and allows in principle, through numerical lattice simulations or other approximation techniques, non-perturbative evaluations of the potential or of the effective coupling constant.     

Equivalence of the Hawking temperature in conformal frames

The conformal invariance of the Hawking temperature, conjectured for the asymptotically flat and stationary black holes by Jacobson and Kang, is semiclassically evaluated for a simple particular case of symmetrical spherically and non-asymptotically flat black hole. By using the Bogoliubov coefficients, the metric euclideanization, the reflection coefficient and the gravitational anomaly, as methods of calculating the Hawking temperature, we find that it is invariant under a specific conformal transformation of the metric. We briefly discuss the results for each method.     

Conformal off-mass-shell extension and elimination of conformal anomalies in quantum gravity

Radiative corrections in the Einstein quantum gravity are made manifestly conformally invariant without changing the S matrix. The conformally invariant form of the classical gravitational action is restored. It is shown, that conformal anomalies, discovered in gravitating systems, do not affect the S matrix. Off the mass shell these anomalies are eliminated by the appropriate choice of a regularization.