The Post-Minkowskian Limit of <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>)-gravity

We formally discuss the post-Minkowskian limit of f(R)-gravity without adopting conformal transformations but developing all the calculations in the original Jordan frame. It is shown that such an approach gives rise, in general, together with the standard massless graviton, to massive scalar modes whose masses are directly related to the analytic parameters of the theory. In this sense, the presence of massless gravitons only is a peculiar feature of General Relativity. This fact is never stressed enough and could have dramatic consequences in detection of gravitational waves. Finally the role of curvature stress-energy tensor of f(R)-gravity is discussed showing that it generalizes the so called Landau-Lifshitz tensor of General Relativity. The further degrees of freedom, giving rise to the massive modes, are directly related to the structure of such a tensor.     

Boundary term in metric <Emphasis Type="Italic">f</Emphasis> (<Emphasis Type="Italic">R</Emphasis>) gravity: field equations in the metric formalism

The main goal of this paper is to get in a straightforward form the field equations in metric f (R) gravity, using elementary variational principles and adding a boundary term in the action, instead of the usual treatment in an equivalent scalar–tensor approach. We start with a brief review of the Einstein–Hilbert action, together with the Gibbons–York–Hawking boundary term, which is mentioned in some literature, but is generally missing. Next we present in detail the field equations in metric f (R) gravity, including the discussion about boundaries, and we compare with the Gibbons–York–Hawking term in General Relativity. We notice that this boundary term is necessary in order to have a well defined extremal action principle under metric variation.     

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.     

The relation between metric and spin-2 formulations of linearized Einstein theory

A twenty-dimensional space of charged solutions of spin-2 equations is proposed. The relation with extended (via dilatation) Poincaré group is analyzed. Locally, each solution of the theory may be described in terms of a potential, which can be interpreted as a metric tensor satisfying linearized Einstein equations. Globally, the nonsingular metric tensor exists if and only if 10 among the above 20 charges do vanish. The situation is analogous to that in classical electrodynamics, where vanishing of magnetic monopole implies the global existence of the electromagnetic potentials. The notion ofasymptotic conformal Yano-Killing tensor is defined and used as a basic concept to introduce an inertial frame in General Relativity via asymptotic conditions at spatial infinity. The introduced class of asymptotically flat solutions is free of supertranslation ambiguities.     

Entropy of the Universe

After a discussion on several limiting cases where General Relativity turns into less sophisticated theories, we find that in the correct thermodynamical and cosmological weak field limit of Einstein’s field equations the entropy of the Universe is R ^3/2-dependent, where R stands for the radius of the causally related Universe. Thus, entropy grows in the Universe, contrary to Standard Cosmology prediction.     

On the phenomenological three-graviton vertex

In General Relativity, the graviton interacts in three-graviton vertex with a tensor that is not the energy-momentum tensor of the gravitational field. We consider the possibility that the graviton interacts with the definite gravitational energy-momentum tensor that we previously found in the G ² approximation. This tensor in a gauge, where nonphysical degrees of freedom do not contribute, is remarkable, because it gives positive gravitational energy density for the Newtonian center in the same manner as the electromagnetic energy-momentum tensor does for the Coulomb center. We show that the assumed three-graviton vertex does not lead to contradiction with the precession of Mercury’s perihelion. In the S-matrix approach used here, the external gravitational field has only a subsidiary role, similar to the external field in quantum electrodynamics. This approach with the assumed vertex leads to the gravitational field that cannot be obtained from a consistent gravity equation.     

Archaic Universe and Cosmological Model: “Big-Bang” as Nucleation by Vacuum

In this work, we examine in depth the physical aspects of the archaic universe described by Euclidean 5-sphere geometry, by using Projective Relativity techniques. We hypothesize that the expansion of the Universe was “ignited” by primordial R processes, and that the big bang consisted of a spatially extended nucleation process which took place at the end of a pre-cosmic phase, characterized by the evolution parameter x₀\underline{x}_{0}. This parameter, which can be considered a quantum precursor of ordinary physical time, is a coordinate of Euclidean 5-sphere metrics. It is so possible to avoid many of the difficulties with standard model and to get rid of ad hoc assumptions. A complete solution to Projective General Relativity (PGR) equations is provided, so as to establish univocal relations between the scale factor R(τ) and cosmic time τ. In this way, the physics and geometry of the cosmological model are specified completely.     

Spiral Galaxy Rotation Curves Determined from Carmelian General Relativity

Equations of motion, in cylindrical co-ordinates, for the observed rotation of gases within the gravitational potential of spiral galaxies have been derived from Carmeli's Cosmological General Relativity theory. A Tully-Fisher type relation results and rotation curves are reproduced without the need for non-baryonic halo dark matter. Two acceleration regimes are discovered that are separated by a critical acceleration m s⁻². For accelerations larger than the critical value the Newtonian force law applies, but for accelerations less than the critical value the Carmelian regime applies. In the Newtonian regime the accelerations fall off as r ⁻², but in the Carmelian regime the accelerations fall off as r ⁻¹. This is new physics but is exactly what is suggested by Milgrom's phenomenological MOND theory.     

Republication of: Relativistic cosmology

This is a republication of a paper by G.F.R. Ellis first published in Proceedings of the International School of Physics: General Relativity and Cosmology, 1971, in which he formulated the framework for relativistic cosmology with an arbitrary background geometry. The article has been selected for publication in the Golden Oldies series of General Relativity and Gravitation. The paper is accompanied by a Golden Oldie Editorial comprising an editorial note written by Bill Stoeger and Ellis’ brief autobiography. Original paper: G. F. R. Ellis: Relativistic cosmology, In: Sachs, R.K. (ed.) Proceedings of the International School of Physics “Enrico Fermi”, Course 47: General relativity and cosmology, pp. 104–182. Academic Press, New York and London (1971). Reprinted with the kind permission of Elsevier Ltd, the current owner of the Academic Press copyright, and of the author.     

Electrodynamics,Gravity and the Corresponding Short-Range Fermi Forces

We use a field theoretical approach to describe the exact General Relativity via a rank-two symmetric tensor field ϕ^μv. We examine the hypothesis that the long range gravitational field has a local counterpart mediated by massive spin-2 bosons which preserves the SU(2) × U(1) gauge symmetry of Electroweak interaction.     

Stability of general relativistic static thick disks: the isotropic Schwarzschild thick disk

We study the stability of general relativistic static thick disks. As an application we consider the thick disk generated by applying the “displace, cut, fill and reflect” method, usually known as the image method, to the Schwarzschild metric in isotropic coordinates. The isotropic Schwarzschild thick disk obtained from this method is the simplest model to describe, in the context of General Relativity, real thick galaxies. Stability under a general first order perturbation of the disk energy momentum tensor is investigated. The first order perturbation, when applied to the conservation equations, leads to a set of differential equations that have fewer equations than unknowns. In this article we search for perturbations in which the perturbation of the four velocity in a certain direction leads to a pressure perturbation in the same direction. We found that, in general, the isotropic Schwarzschild thick disk is stable under these kinds of perturbations.     

Poincaré Duality and <Emphasis Type="Italic">G</Emphasis><Superscript>+++</Superscript> Algebras

Theories with General Relativity as a sub-sector exhibit enhanced symmetries upon dimensional reduction, which is suggestive of “exotic dualities”. Upon inclusion of time-like directions in the reductions one can dualize to theories in different space-time signatures. We clarify the nature of these dualities and show that they are well captured by the properties of infinite-dimensional symmetry algebras (G ⁺⁺⁺- algebras), but only after taking into account that the realization of Poincaré duality leads to restrictions on the denominator subalgebra appearing in the non-linear realization. The correct realization of Poincaré duality can be encoded in a simple algebraic constraint, that is invariant under the Weyl-group of the G ⁺⁺⁺-algebra, and therefore independent of the detailed realization of the theory under consideration. We also construct other Weyl-invariant quantities that can be used to extract information from the G ⁺⁺⁺-algebra without fixing a level decomposition. Post-doctoraal onderzoeker van het Fonds voor Wetenschappelijk Onderzoek, Vlaanderen.     

Geometric Equations of State in Friedmann-Lemaître Universes Admitting Matter and Ricci Collineations

As a rule in General Relativity the spacetime metric fixes the Einstein tensor and through the Field Equations (FE) the energy-momentum tensor. However one cannot write the FE explicitly until a class of observers has been considered. Every class of observers defines a decomposition of the energy-momentum tensor in terms of the dynamical variables energy density (), the isotropic pressure (p), the heat flux q ^a and the traceless anisotropic pressure tensor _ab . The solution of the FE requires additional assumptions among the dynamical variables known with the generic name equations of state. These imply that the properties of the matter for a given class of observers depends not only on the energy-momentum tensor but on extra a priori assumptions which are relevant to that particular class of observers. This makes difficult the comparison of the physics observed by different classes of observers for the same spacetime metric. One way to overcome this unsatisfactory situation is to define the extra condition required among the dynamical variables by a geometric condition, which will be based on the metric and not to the observers. Among the possible and multiple conditions one could use the consideration of collineations. We examine this possibility for the Friedmann-Lemaître-Robertson-Walker models admitting matter and Ricci collineations and determine the equations of state for the comoving observers. We find linear and non-linear equations of state, which lead to solutions satisfying the energy conditions, therefore describing physically viable cosmological models.     

On Superluminal Particles and the Extended Relativity Theories

Superluminal particles are studied within the framework of the Extended Relativity theory in Clifford spaces (C-spaces). In the simplest scenario, it is found that it is the contribution of the Clifford scalar component π of the poly-vector-valued momentum which is responsible for the superluminal behavior in ordinary spacetime due to the fact that the effective mass is imaginary (tachyonic). However, from the point of view of C-space, there is no superluminal (tachyonic) behavior because the true physical mass still obeys M ²>0. Therefore, there are no violations of the Clifford-extended Lorentz invariance and the extended Relativity principle in C-spaces. It is also explained why the charged muons (leptons) are subluminal while its chargeless neutrinos may admit superluminal propagation. A Born’s Reciprocal Relativity theory in Phase Spaces leads to modified dispersion relations involving both coordinates and momenta, and whose truncations furnish Lorentz-violating dispersion relations which appear in Finsler Geometry, rainbow-metrics models and Double (deformed) Special Relativity. These models also admit superluminal particles. A numerical analysis based on the recent OPERA experimental findings on alleged superluminal muon neutrinos is made. For the average muon neutrino energy of 17 GeV, we find a value for the magnitude that, coincidentally, is close to the mass of the muon m _μ =105.7 MeV.     

Towards a general solution of the Hamiltonian constraints of General Relativity

The present work has a double aim. On the one hand, we call attention on the relationship existing between the Ashtekar formalism and other gauge-theoretical approaches to gravity, in particular the Poincaré Gauge Theory. On the other hand, we study two kinds of solutions for the constraints of General Relativity, consisting of two mutually independent parts, namely a general three-metric-dependent contribution to the extrinsic curvature K _ab in terms of the Cotton–York tensor, and besides it further metric independent contributions, which we analyze in particular in the presence of isotropic three-metrics.     

Magnetized Bianchi Type III String Universe with Time Decaying Vacuum Energy Density Λ

Exact solution of Einstein’s field equations is obtained for massive string cosmological model of Bianchi III space-time using the technique given by Letelier (Phys. Rev. D 28:2414, 1983) in presence of perfect fluid and decaying vacuum energy density Λ. To get the deterministic solution of the field equations the expansion θ in the model is considered as proportional to the eigen value s² ₂\sigma^{2}_{~2} of the shear tensor s^j _i\sigma^{j}_{~i} and also the fluid obeys the barotropic equation of state. It is observed that the particle density and the tension density of the string are comparable at the two ends and they fall off asymptotically at similar rate. But in early stage as well as at the late time of the evolution of the universe we have two types of scenario (i) universe is dominated by massive strings and (ii) universe is dominated by strings depending on the nature of the two constants L and ℓ. The value of cosmological constant Λ for the model is found to be small and positive which is supported by the results from recent supernovae Ia observations. Some physical and geometric properties of the model are also discussed.     

The problem of stability of the homogeneous and isotropic universe in the relativistic theory of gravitation

The problem of stability of the homogeneous and isotropic Universe with respect to small perturbations in the gravitational field and matter characteristics is studied in the framework of the relativistic theory of gravitation. The equations for small perturbations of the metric tensor g _μν, energy density ρ, and pressure p are obtained in the linear approximation. The solutions to these equations are found when perturbations depend only on time. The physical character of the obtained solutions is analyzed. A comparison with the results of General Relativity yields the conclusion that all differences are due to the graviton mass.     

The quadratic Lagrangians in General Relativity

The solutions of the General Relativity equations with quadratic LagrangiansR _iklmR^iklm, R_ikR^ik, R² are studied. It is shown that nontrivial Euclidian (atr ) solution of the theory equations does not exist whenT0 (T is a trace of the energy-momentum tensor of matter). The Schwarzschild solution is not an external part of a total solution whenT0. Under conditionT=R=0 LagrangiansR _iklmR^iklm, R_ikR^ik lead to the identical field equations, so there exist the only quadratic Lagrangian and the only field equations. This equation has a solution with an external part being a standard Schwarzschild solution for the statical spherically symmetric case.