A. Stabile

Hydrostatic equilibrium and stellar structure in f ( R ) gravity

Dipartimento di Ingegneria, Universita’ del Sannio,Palazzo Dell’Aquila Bosco Lucarelli, Corso Garibaldi, 107 - 82100, Benevento, Italy.(Dated: January 4, 2011)We investigate the hydrostatic equilibrium of stellar structure by taking into account the modi-fied Lan´e-Emden equation coming out from f(R)-gravity. Such an equation is obtained in metricapproach by considering the Newtonian limit of f(R)-gravity, which gives rise to a modified Poissonequation, and then introducing a relation between pressure and density with polytropic index n. Themodified equation results an integro-differential equation, which, in the limit f(R) → R, becomesthe standard Lan´e-Emden equation. We find the radial profiles of gravitational potential by solvingfor some values of n. The comparison of solutions with those coming from General Relativity showsthat they are compatible and physically relevant.

Spherical symmetry in f(R)-gravity

Spherical symmetry in f(R)-gravity is discussed in detail considering also the relations to the weak field limit. Exact solutions are obtained for constant Ricci curvature scalar and for Ricci scalar depending on the radial coordinate. In particular, we discuss how to obtain results which can be consistently compared with general relativity giving the well known post-Newtonian and post-Minkowskian limits. Furthermore, we implement a perturbation approach to obtain solutions up to the first order starting from spherically symmetric backgrounds. Exact solutions are given for several classes of f(R)-theories in both R= constant and R = R(r).

Axially symmetric solutions in f(R)-gravity

Axially symmetric solutions for f(R)-gravity can be derived starting from exact spherically symmetric solutions achieved by Noether symmetries. The method takes advantage of a complex coordinate transformation previously developed by Newman and Janis in general relativity. An example is worked out to show the general validity of the approach. The physical properties of the solution are also considered.

Fourth order gravity and experimental constraints on Eddington parameters

1. General Relativity (GR) is the cornerstone theory among the several attempts proposed to describe gravity. It represents an elegant approach furnishing several phenomenological predictions and its validity, in the Newtonian limit regime, is experimentally probed up to the Solar System scales. However, also at these scales, some conundrums come out as the indications of an apparent, anomalous, long–range acceleration revealed from the data analysis of Pioneer 10/11, Galileo, and Ulysses spacecrafts which are difficult to be framed in the standard scheme of GR and its low energy limit [1, 2]. Furthermore, at galactic distances, huge bulks of dark matter are needed to provide realistic models matching with observations. In this case, retaining GR and its low energy limit implies the introduction of an actually unknown ingredient. We face a similar situation even at larger scales: clusters of galaxies are gravitationally stable and bound only if large amounts of dark matter are supposed in their potential wells. Finally, an unknown form of dark energy is required to explain the observed accelerated expansion of cosmic fluid. Summarizing, almost 95% of matter-energy content of the universe is unknown in the framework of Standard Cosmological Model while we can experimentally probe only gravity and ordinary (baryonic and radiation) matter. Considering another point of view, anomalous acceleration (Solar System), dark matter (galaxies and galaxy clusters), dark energy (cosmology) could be nothing else but the indications that shortcomings are present in GR and gravity is an interaction depending on the scale. The assumption of a linear Lagrangian density in the Ricci scalar R for the Hilbert-Einstein action could be too simple to describe gravity at any scale and more general approaches should be pursued to match observations. Among these schemes, several motivations suggest to generalize GR by considering gravitational actions where generic functions of curvature invariants are present. Specifically, actions of the form

A general solution in the Newtonian limit of f(R) - gravity

We show that any analytic f(R)-gravity model, in the metric approach, presents a weak field limit where the standard Newtonian potential is corrected by a Yukawa-like term. This general result has never been pointed out but often derived for some particular theories. This means that only f(R) = R allows to recover the standard Newton potential while this is not the case for other relativistic theories of gravity. Some considerations on the physical consequences of such a general solution are addressed.

The Newtonian limit of metric gravity theories with quadratic Lagrangians

The Newtonian limit of fourth-order gravity is worked out discussing its viability with respect to the standard results of general relativity. We investigate the limit in the metric approach which, with respect to the Palatini formulation, has been much less studied in the recent literature, due to the higher order of the field equations. In addition, we refrain from exploiting the formal equivalence of higher-order theories considering the analogy with specific scalar–tensor theories, i.e. we work in the so-called Jordan frame in order to avoid possible misleading interpretations of the results. Explicit solutions are provided for several different types of Lagrangians containing powers of the Ricci scalar as well as combinations of the other curvature invariants. In particular, we develop the Greens function method for fourth-order theories in order to find out solutions. Finally, the consistency of the results with respect to general relativity is discussed.

Conformal Transformations and Weak Field Limit of Scalar-Tensor Gravity

(Dated: December 5, 2013)The weak field limit of scalar tensor theories of gravity is discussed in view of conformal trans-formations. Specifically, we consider how physical quantities, like gravitational potentials derivedin the Newtonian approximation for the same scalar-tensor theory, behave in the Jordan and in theEinstein frame. The approach allows to discriminate features that are invariant under conformaltransformations and gives contributions in the debate of selecting the true physical frame. As aparticular example, the case of f(R) gravity is considered.

The Post-Minkowskian Limit of f(R)-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.

Self-Gravitating Systems in Extended Gravity

Starting from the weak field limit, we discuss astrophysical applications of Extended Theories of Gravity where higher order curvature invariants and scalar fields are considered by generalizing the Hilbert-Einstein action linear in the Ricci curvature scalar R. Results are compared to General Relativity in the hypothesis that Dark Matter contributions to the dynamics can be neglected thanks to modified gravity. In particular, we consider stellar hydrostatic equilibrium, galactic rotation curves, and gravitational lensing. Finally, we discuss the weak field limit in the Jordan and Einstein frames pointing out how effective quantities, as gravitational potentials, transform from one frame to the other and the interpretation of results can completely change accordingly.