Conformal transformations and conformal invariance in gravitation

Journal of Cosmology and Astroparticle Physics

The R + R 2 model is successful in describing inflation, as it provides an excellent fit to the full set of available observational data. On the other hand, the same model is the simplest extension of general relativity which does not produce higher derivative ghosts and related instabilities. Long ago, it was proposed to treat all terms which cause higher derivative instabilities as small perturbations that could avoid the presence of ghosts in the spectrum. We put this proposal into practice and consider an explicit example of treating more complicated higher derivative terms as small perturbations over the R + R 2 model by introducing the R R term into the action. Within the described scheme, it is possible to obtain an upper bound on the coefficient of this non-scale-free sixth-derivative term by mapping the theory into a one-scalar field potential. It is shown that the result differs from treating this term on equal footing with other terms that requires mapping to a two-scalar field model, and in general leads to different observational consequences. Contents 1 Introduction 1 2 The R + R 2 Model 3 3 Treating R R term as a small perturbation to the R + R 2 model 4 3.1 Slow-roll conditions 8 4 Conclusions 13 A Action in terms of scalar fields 13 A.1 Weak field approximation 14

Physical Review D

We study static spherically and hyperbolically symmetric solutions of the Einstein equations in the presence of a conformally coupled scalar field and compare them with those in the space filled with a minimally coupled scalar field. We then study the Kantowski-Sachs cosmological solutions, which are connected with the static solutions by the duality relations. The main ingredient of these relations is an exchange of roles between the radial and the temporal coordinates, combined with the exchange between the spherical and hyperbolical two-dimensional geometries. A brief discussion of questions such as the relation between the Jordan and the Einstein frames and the description of the singularity crossing is also presented.

General Relativity and Gravitation, 2010

The 1+3 covariant approach and the covariant gauge-invariant approach to perturbations are used to analyze in depth conformal transformations in cosmology. Such techniques allow us to obtain insights on the physical meaning of these transformations when applied to non-standard gravity. The results obtained lead to a number of general conclusions on the change of some key quantities describing any two conformally related cosmological models. For example, even if some of the geometrical properties of the cosmology are preserved (homogeneous and isotropic Universes are mapped into homogeneous and isotropic universes), it can happen that decelerating cosmologies can be mapped into accelerated ones. From the point of view of the cosmological perturbations it is shown how these fluctuation transform. We find that first-order vector and tensor perturbations equations are left unchanged in their structure by the conformal transformation, but this cannot be said of the scalar perturbations, which present differences in their evolutionary features. The results obtained are then explicitly interpreted and verified with the help of some clarifying examples based on f (R)-gravity cosmologies. Keywords conformal transformations • modified gravity • cosmology • 1+3 covariant approach • covariant gauge invariant theory of perturbations

Advances in High Energy Physics, 2015

Metric-affine geometry provides a nontrivial extension of the general relativity where the metric and connection are treated as the two independent fundamental quantities in constructing the spacetime (with nonvanishing torsion and nonmetricity). In this paper, we study the generic form of action in this formalism and then construct the Weyl-invariant version of this theory. It is shown that, in Weitzenböck space, the obtained Weyl-invariant action can cover the conformally invariant teleparallel action. Finally, the related field equations are obtained in the general case.

Classical and Quantum Gravity

The conformal transformation of the Hawking-Hayward quasi-local mass is reexamined. It has been found recently that the conformal transformation of the latter exhibits the 'wrong' conformal factor compared to the way usual masses transform under conformal transformations of spacetime. We show, in analogy with what was found recently for the Misner-Sharp mass, that unlike the purely geometric definition of the Hawking-Hayward mass, the latter exhibits the 'right' conformal factor whenever expressed in terms of its material content via the field equations. The case of conformally invariant scalar-tensor theories of gravity is also examined. The equivalence between the Misner-Sharp mass and the Hawking-Hayward mass for spherically symmetric spacetimes manifests itself by giving identical peculiar behaviors under conformal transformations.

International Journal of Modern Physics D

The conformal transformation of the Misner–Sharp mass is reexamined. It has recently been found that this mass does not transform like usual masses do under conformal mappings of spacetime. We show that when it comes to conformal transformations, the widely used geometric definition of the Misner–Sharp mass is fundamentally different from the original conception of the latter. Indeed, when working within the full hydrodynamic setup that gave rise to that mass, i.e. the physics of gravitational collapse, the familiar conformal transformation of a usual mass is recovered. The case of scalar–tensor theories of gravity is also examined.

Physical Review D

We show that the violation of the null energy condition by matter, required by traversable wormholes, can be removed in spacetimes with torsion. In addition, we show that this violation can also be removed in the conformal frame obtained by a Weyl transformation. This comes about because both conformally transformed wormholes and wormholes in spacetimes with torsion become sustained thanks to a combination of "normal" matter, that satisfies the null energy condition, and an induced "geometric" energy-momentum tensor that is able to greatly violate the null energy condition.

Physical Review D

Physical Review D

The Noether charge associated to diffeomorphism invariance in teleparallel gravity is derived. It is shown that the latter yields the ADM mass of an asymptotically flat spacetime. The black hole entropy is then investigated based on Wald's prescription that relies on the Noether charge. It is shown that, like in general relativity, the surface gravity can be factored out from such a charge. Consequently, the similarity with the first law of thermodynamics implied by such an approach in general relativity does show up also in teleparallel gravity. It is found that, based on the expression of the first law of black hole mechanics, which is preserved in teleparallel gravity, entropy can thus be extracted from such a Noether charge. The resulting entropy can very naturally be expressed as a volume integral, though. As such, it is shown that the conformal issue that plagues the entropy-area law within general relativity does not arise in teleparallel gravity based on Wald's approach. The physics behind these features is discussed.

Classical and Quantum Gravity

No. In a number of papers Green and Wald argue that the standard FLRW model approximates our Universe extremely well on all scales, except close to strong field astrophysical objects. In particular, they argue that the effect of inhomogeneities on average properties of the Universe (backreaction) is irrelevant. We show that this latter claim is not valid. Specifically, we demonstrate, referring to their recent review paper, that (i) their two-dimensional example used to illustrate the fitting problem differs from the actual problem in important respects, and it assumes what is to be proven; (ii) the proof of the trace-free property of backreaction is unphysical and the theorem about it fails to be a mathematically general statement; (iii) the scheme that underlies the trace-free theorem does not involve averaging and therefore does not capture crucial non-local effects; (iv) their arguments are to a large extent coordinate-dependent, and (v) many of their criticisms of backreaction frameworks do not apply to the published definitions of these frameworks. It is No proof that backreaction is irrelevant 2 therefore incorrect to infer that Green and Wald have proven a general result that addresses the essential physical questions of backreaction in cosmology.

Journal of Physics: Conference Series

The scalar-tensor gravity as one of the alternative theories of general relativity has some interesting features. In the case of compact objects it predicts a unique phenomenon which is known as spontaneous scalarization, as shown by Damour and Espasito-Farese [1]. This phenomenon does not appear within the general relativity. In this note, we investigate a new conformal coupling function i.e., A(φ) = 1 − ln[1 − 0.5β(φ − φ 0)2] and its consequences in neutron stars. In the case of a polytropic equation of state, we have found that the spontaneous scalarization starts to appear when βc ≈ −4.35. We also found that for the realistic equation of states the βc depends on the composition and the stiffness of the equation of state.

The European Physical Journal C

We classify singularities in FRW cosmologies, which dynamics can be reduced to the dynamical system of the Newtonian type. This classification is performed in terms of the geometry of a potential function if it has poles. At the sewn singularity, which is of a finite scale factor type, the singularity in the past meets the singularity in the future. We show that such singularities appear in the Starobinsky model in f (R) =R + γR 2 in the Palatini formalism, when dynamics is determined by the corresponding piecewise-smooth dynamical system. As an effect we obtain a degenerate singularity. Analytical calculations are given for the cosmological model with matter and the cosmological constant. The dynamics of model is also studied using dynamical system methods. From the phase portraits we find generic evolutionary scenarios of the evolution of the universe. For this model, the best fit value of γ = 3γ H 2 0 is equal 9.70 × 10 −11. We consider a model in both Jordan and Einstein frames. We show that after transition to the Einstein frame we obtain both the form of the potential of the scalar field and the decaying Lambda term. Capozziello for encouraging us to consider this model in different frames. The work has been supported by the Polish National Science Centre (NCN), project DEC-2013/09/B/ ST2/03455.

The European Physical Journal C

We consider FRW cosmology in f (R) = R + γ R 2 + δ R 3 modified framework. The Palatini approach reduces its dynamics to the simple generalization of Friedmann equation. Thus we study the dynamics in twodimensional phase space with some details. After reformulation of the model in the Einstein frame, it reduces to the FRW cosmological model with a homogeneous scalar field and vanishing kinetic energy term. This potential determines the running cosmological constant term as a function of the Ricci scalar. As a result we obtain the emergent dark energy parametrization from the covariant theory. We study also singularities of the model and demonstrate that in the Einstein frame some undesirable singularities disappear.

The European Physical Journal Plus

A concise description is presented in the basic features of the formalism of noncanonical spacetime volume forms and its application in modified gravity theories and cosmology. The well-known unimodular gravity theory appears as a very special case. Concerning the hot issues facing cosmology now, we specifically briefly outline the construction of: (a) unified description of dark energy and dark matter as manifestations of a single material entity-a special scalar field "darkon"; (b) quintessential models of universe evolution with a gravity "inflaton"-assisted dynamical Higgs mechanism-dynamical suppression/generation of spontaneous electroweak gauge symmetry breaking in the "early"/"late" universe; (c) unification of dark energy and dark matter with diffusive interaction among them; and (d) mechanism for suppression of fifth force without fine-tuning. 1 Non-Riemannian volume-form formalism A broad class of actively developed modified/extended gravitational theories is based on employing alternative non-Riemannian spacetime volume forms (metric-independent generally covariant volume elements) in the pertinent Lagrangian actions instead of, or alongside with, the canonical Riemannian volume element given by the square root of the determinant of the Riemannian metric. This method was originally proposed in [1,2] and for a concise geometric formulation using differential forms combined with canonical Hamiltonian formalism for systems with constraints (gauge symmetries), see [3,4] (an earlier geometric formulation with a "quartet" of scalar fields appeared in [5]). Volume forms are fairly basic objects in differential geometry-they exist on arbitrary differentiable manifolds and define covariant (under general coordinate reparameterizations) integration measures.

Modern Physics Letters A, 2015

We present black hole solutions in (2+1)-dimensional Einstein’s theory of gravity coupled with Born–Infeld (BI) nonlinear electrodynamic and a massless self-interacting scalar field. The model has five free parameters: mass [Formula: see text], cosmological constant [Formula: see text], electric [Formula: see text] and scalar [Formula: see text] charges and BI parameter [Formula: see text]. To attain exact solution for such a highly nonlinear system we adjust, i.e. finely tune, the parameters of the theory with the integration constants. In the limit [Formula: see text], we recover the results of Einstein–Maxwell–Scalar theory, obtained before. The self-interacting potential admits finite minima apt for the vacuum contribution. Hawking temperature of the model is investigated versus properly tuned parameters. By employing this tuned-solution as basis, we obtain also a dynamic solution which in the proper limit admits the known solution in Einstein gravity coupled with self-interacti...

Journal of Cosmology and Astroparticle Physics

A 'novel' pure theory of Einstein-Gauss-Bonnet gravity in four-spacetime dimensions can be constructed by rescaling the Gauss-Bonnet coupling constant, seemingly eluding Lovelock's theorem. Recently, however, the well-posedness of this model has been called into question. Here we apply a 'dimensional regularization' technique, first used by Mann and Ross to write down a D → 2 limit of general relativity, to the case of pure Einstein-Gauss-Bonnet gravity. The resulting four-dimensional action is a particular Horndeski theory of gravity matching the result found via a Kaluza-Klein reduction over a flat internal space. Some cosmological solutions of this four-dimensional theory are examined. We further adapt the technique to higher curvature Lovelock theories of gravity, as well as a low-energy effective string action with an α ′ correction. With respect to the D → 4 limit of the α ′-corrected string action, we find we must also rescale the dilaton to have a non-singular action in four dimensions. Interestingly, when the conformal rescaling Φ is interpreted as another dilaton, the regularized string action appears to be a special case of a covariant multi-Galileon theory of gravity.

Physical Review D, 2015

In this paper we investigate the chameleon effect in the different conformal frames of the Brans-Dicke theory. Given that, in the standard literature on the subject, the chameleon is described in the Einstein frame almost exclusively, here we pay special attention to the description of this effect in the Jordan and in the string frames. It is shown that, in general, terrestrial and solar system bounds on the mass of the BD scalar field, and bounds of cosmological origin, are difficult to reconcile at once through a single chameleon potential. We point out that, in a cosmological context, provided that the effective chameleon potential has a minimum within a region of constant density of matter, the Brans-Dicke theory transmutes into general relativity with a cosmological constant, in that region. This result, however, can be only locally valid. In cosmological settings de Sitter-general relativity is a global attractor of the Brans-Dicke theory only for the quadratic potential V (φ) = M 2 φ 2 , or for potentials that asymptote to M 2 φ 2 .

Physical Review D, 2019

Frontiers in Physics

The dynamics of the scalar field and particle in a conformal frame are considered. The conformal Klein-Gordon equation describing the scalar field is transformed into the quantum Telegraph equation in Minkowski space-time. The conformal factor acts like a background field having a perfect energy-momentum tensor. The scalar field decays exponentially with time during inflation allowing the conformal field to induce space energy. The conformal field grows with time at the expense of decreasing the energy density of the real scalar field. Einstein’s tensor embodies an energy-momentum tensor representing the background contribution reflecting the matter aspect of the gravitational field. The energy density arising from the conformal field is negative. The background energy associated with Einstein’s curvature tensor gives rise to massive gravitons that act like a cosmological constant. In an expanding Universe, this particular case yields a background energy proportional to the square o...

Journal of High Energy Physics, 2011

We study spin-2 excitations in the background of the recently-discovered type-IIB solutions of D'Hoker et al. These are holographically-dual to defect conformal field theories, and they are also of interest in the context of the Karch-Randall proposal for a string-theory embedding of localized gravity. We first generalize an argument by Csaki et al to show that for any solution with four-dimensional anti-de Sitter, Poincaré or de Sitter invariance the spin-2 excitations obey the massless scalar wave equation in ten dimensions. For the interface solutions at hand this reduces to a Laplace-Beltrami equation on a Riemann surface with disk topology, and in the simplest case of the supersymmetric Janus solution it further reduces to an ordinary differential equation known as Heun's equation. We solve this equation numerically, and exhibit the spectrum as a function of the dilaton-jump parameter ∆φ. In the limit of large ∆φ a nearly-flat linear-dilaton dimension grows large, and the Janus geometry becomes effectively five-dimensional. We also discuss the difficulties of localizing four-dimensional gravity in the more general backgrounds with NS5-brane or D5-brane charge, which will be analyzed in detail in a companion paper.

Physical review, 2017

The well known working example which Donoghue quotes is the non-linear sigma model of mesons, which is valid significantly below the scale Λ QCD , the scale of confinement at which QCD strongly couples and below which every perturbative calculation ceases to be valid. 2 Some inflationary models comprise more scalar fields. Our considerations are easily generalized to encompass these more general cases. Similarly, our considerations can be generalized to include operators that include other (fermionic and vector) matter fields. 3 When an internal symmetry is imposed on the classical level, it typically also survives at the quantum level, which is what we assume is the case in this work. Notable exceptions are spontaneous symmetry breaking by a Higgs-like condensate and 'breaking' of gauge symmetry via quantum effects such as in the scalar electrodynamics on de Sitter space [7, 8].