Vacuum decay and quadratic gravity: the massive case

General Relativity and Gravitation - In this article we discuss some aspects of double field theory cosmology with an emphasis on the role played by the dilaton. The cosmological solutions of...     

On the cosmological constant problem and brane-world geometry

The cosmological constant problem is examined within the context of the covariant brane-world gravity, based on Nash’s embedding theorem for Riemannian geometries. We show that the vacuum structure of the brane-world is more complex than General Relativity’s because it involves extrinsic elements, in specific, the extrinsic curvature. In other words, the shape (or local curvature) of an object becomes a relative concept, instead of the “absolute shape” of General Relativity. We point out that the immediate consequence is that the cosmological constant and the energy density of the vacuum quantum fluctuations have different physical meanings: while the vacuum energy density remains confined to the four-dimensional brane-world, the cosmological constant is a property of the bulk’s gravitational field that leads to the conclusion that these quantities cannot be compared, as it is usually done in General Relativity. Instead, the vacuum energy density contributes to the extrinsic curvature, which in turn generates Nash’s perturbation of the gravitational field. On the other hand, the cosmological constant problem ceases to be in the brane-world geometry, reappearing only in the limit where the extrinsic curvature vanishes.     

Letter: Charging Symmetries and Linearizing Potentials for Gravity Models with Symplectic Symmetry

In this letter we continue to study a class offour-dimensional gravity models with n Abelian vectorfields and Sp(2n, R)/U(n) coset of scalar fields. Thisclass contains General Relativity (n = 0) andEinstein-Maxwell dilaton axion theory (n = 1),which arises in the low energy limit of heterotic stringtheory. We perform reduction of the model with arbitraryn to three dimensions and study the subgroupof non-gauge symmetries of the resultingtheory. First, we find an explicit form these symmetriesusing the Ernst matrix potential formulation. Second, weconstruct a new matrix variable which linearlytransforms under the action of the non-gaugetransformations. Finally, we establish one generalinvariant of the non gauge symmetry subgroup, whichallows us to clarify this subgroup structure.     

General relativity and the schwarzschild singularity

The discrepancy in General Relativity between its general statements and the vacuum singular solutions of Einstein’s equations is analyzed.     

Freely Falling 2-Surfaces and the Effective Gravitational Mass

We derive an expression for the effectivegravitational mass for any closed spacelike 2-surface.This effective gravitational energy is defined directlythrough the geometrical quantity of the freely falling 2-surface and thus is well adapted to intuitiveexpectation that the gravitational mass should bedetermined by the motion of a test body moving freely inthe gravitational field. We find that this effective gravitational mass has a reasonable positivevalue for a small sphere in the non-vacuum space-timesand can be negative for the vacuum case. Further, thiseffective gravitational energy is compared with the quasi-local energy based on the (2 + 2)formalism of General Relativity. Although some gaugefreedoms exist, analytic expressions of the quasi-localenergy for vacuum cases are the same as the effective gravitational mass. Especially, we see that thecontribution from the cosmological constant is the samein general cases.     

Letter: Gravitational Potential in the Palatini Formulation of Modified Gravity

General Relativity has so far passed almost all the ground-based and solar-system experiments. Any reasonable extended gravity models should consistently reduce to it at least in the weak field approximation. In this work we derive the gravitational potential for the Palatini formulation of the modified gravity of the L(R) type which admits a de Sitter vacuum solution. We argue that the Newtonian limit is always obtained in those class of models and the deviations from General Relativity are very small for a slowly moving source.     

Weakened field equations in General Relativity admitting an ‘unphysical’ metric

Certain weakened vacuum field equations which have been suggested as alternatives to the vacuum field equations of General Relativity are investigated. It is shown that they each possess a solution which corresponds to an isolated mass at the origin which repels test particles. In view of the contradiction with experiment this solution is an embarrassment to these proposed vacuum field equations.On leave of absence from the Department of Mathematics, The University, Bristol.     

The general relativity with conformal units

General Relativity rewritten in conformal units identifies conformal intervals with the real observational distances. This identification gives a base to explain all epochs of the Universe evolution including Ia supernova luminosity long distance-redshift relation by the dominance of the Casimir vacuum energy of all physical fields. A set of arguments is discussed in favor that SNe Ia data in the conformal units can be an evidence of the conformal twistor structure of the space-time as a nonlinear realization of the affine group, like the nonlinear realization of chiral symmetry and phenomenological Lagrangian is evidence of the quark structure of hadrons.     

Non-spherical sources of static gravitational fields: Investigating the boundaries of the no-hair theorem

A new, globally regular model describing a static, non spherical gravitating object in General Relativity is presented. The model is composed by a vacuum Weyl–Levi-Civita special field – the so called gamma metric – generated by a regular static distribution of mass-energy. Standard requirements of physical reasonableness such as, energy, matching and regularity conditions are satisfied. The model is used as a toy in investigating various issues related to the directional behavior of naked singularities in static spacetimes and the black hole (Schwarzschild) limit.     

The homogeneous gravitational force field in general relativity

The concept of a gravitational force in General Relativity is reintroduced. The theory of space-like congruences is established and is applied to the discussion of the existence of a homogeneous gravitational force-field in curved space-time. It is shown in vacuum (vanishing Ricci tensor) that such a force-field cannot exist.     

Universe as a representation of affine and conformal symmetries

The evolution of the Universe is considered by means of a nonlinear realization of affine and conformal symmetries via Maurer-Cartan forms. Conformal symmetry is realized by the geometry of similarity with the Dirac scalar dilaton. We provide preliminary quantitative evidence that the zeroth harmonic of the Dirac scalar dilaton may lead to observationally viable cosmology, where the type la supernova luminosity distances-redshift relation can be explained by vacuum dilaton dark energy. The diffeo-invariance of spin connection coefficients of the affine formulation leaves only one degree of freedom of strong gravitation waves. We discuss that the dark matter effect in spiral galaxies can be described by the gravitation waves expressed through the spin connection coefficients of the affine formulation. We show that, the evolution equations of the affine gravitons with respect to the dilaton zeroth mode coincide with the equations of “squeezed oscillator”. The list of theoretical and observational arguments is given in favor of that the origin of the Universe can be described by quantum vacuum creation of these squeezed oscillators.     

A Brief Remark on Energy Conditions and the Geroch-Jang Theorem

The status of the geodesic principle in General Relativity has been a topic of some interest in the recent literature on the foundations of spacetime theories. Part of this discussion has focused on the role that a certain energy condition plays in the proof of a theorem due to Bob Geroch and Pong-Soo Jang [“Motion of a Body in General Relativity.” Journal of Mathematical Physics 16(1) (1975)] that can be taken to make precise the claim that the geodesic principle is a theorem, rather than a postulate, of General Relativity. In this brief note, I show, by explicit counterexample, that not only is a weaker energy condition than the one Geroch and Jang state insufficient to prove the theorem, but in fact a condition still stronger than the one that they assume is necessary.     

Scale invariance and spontaneous symmetry breaking

Spontaneous breaking of gauge symmetries is studied in theories with nonlinearly realized scale invariance. The classically sliding vacuum expectation values are fixed through quantum corrections. The anomaly of the dilatation current determines the vacuum energy density as well as the dilaton mass. The coupling of gravity to matter is modified in such a way that the cosmological constant vanishes.     

The Chaotic Mixmaster and the Suppression of Chaos in Scalar–Tensor Cosmologies

We show that there is a threshold in energy for the onset of chaos in cosmology for the Universe described as a dynamical system derived from the Einstein equations of General Relativity (GR). In the case of the mixmaster model (homogeneous and anisotropic cosmology with a Bianchi IX metric), the chaos occurs precisely at the prescribed necessary value H _vac=0 of the GR for the energy of the Universe while the system is found to be regular for H<0 and chaotic for H>0 with respect to its pure vacuum part. In the case of generalized scalar tensor theories within the Bianchi IX model, we show using the ADM formalism and a conformal transformation that the energy of the dynamical system as compared to vacuum lies below the zero energy threshold. The system is thus not exhibiting chaos and the conclusion still holds in the presence of ordinary matter as well. The suppression of chaos occurs in a similar way for stiff matter alone.     

Thermodynamics and cosmology

A new type of cosmological history which includes large-scale entropy production is proposed. These cosmologies are based on a reinterpretation of the matterenergy stress tensor in Einstein's equations. This modifies the usual adiabatic energy conservation laws, thereby leading to a possible irreversible matter creation. This creation corresponds to an irreversible energy flow from the gravitational field to the created matter constituents. This new point of view results from the consideration of thermodynamics of open systems in the framework of cosmology. It appears that the usual initial singularity is structurally unstable with respect to irreversible matter creation. The corresponding cosmological history therefore starts from an instability of the vacuum rather than from a singularity. The universe evolves through an inflationary phase. This appears to be an attractor independent of the initial vacuum fluctuation.This work is reproduced in part from I. Prigogine, J. Geheniau, E. Gunzig, and P. Nardone (1989), Thermodynamics and Cosmology,General Relativity and Gravitation,21, 767.     

The general class of the vacuum spherically symmetric equations of the general relativity theory

The system of the spherical-symmetric vacuum equations of the General Relativity Theory is considered. The general solution to a problem representing two classes of line elements with arbitrary functions g ₀₀ and g ₂₂ is obtained. The properties of the found solutions are analyzed.     

Horizonless,singularity-free,compact shells satisfying NEC

Gravitational collapse singularities are undesirable, yet inevitable to a large extent in General Relativity. When matter satisfying null energy condition (NEC) collapses to the extent a closed trapped surface is formed, a singularity is inevitable according to Penrose’s singularity theorem. Since positive mass vacuum solutions are generally black holes with trapped surfaces inside the event horizon, matter cannot collapse to an arbitrarily small size without generating a singularity. However, in modified theories of gravity where positive mass vacuum solutions are naked singularities with no trapped surfaces, it is reasonable to expect that matter can collapse to an arbitrarily small size without generating a singularity. Here we examine this possibility in the context of a modified theory of gravity with torsion in an extra dimension. We study singularity-free static shell solutions to evaluate the validity of NEC on the shell. We find that with sufficiently high pressure, matter can be collapsed to arbitrarily small size without violating NEC and without producing a singularity.