Gravitazione ed estensioni della Relatività Generale

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Gravitazione ed estensioni della Relatività Generale


Arturo Stabile



Extended theories of gravity at galactic scales

Modifications of the gravitational potential alter the asymptotic Keplerian behaviour usually adopted and, when we consider ETG, a correction emerges and leads to a specific rotation curve. Assuming realistic models of the morphology of the galaxy and relaxing the thin disk assumption lead to interesting models to estimate the speed of rotation. Since the Newtonian limit derives from the linearization of the field equations, one can use the superposition principle and we obtain the effective gravitational potential in the galaxy. By comparing the theoretical results with experimental evidences we can obtain constraints on free parameters of the theory. Furthermore in this case it is possible to quantify the fluctuations in the vertical plane of the galaxy. This is only an initial analysis where a test body is embedded in the gravitational field of the galaxy, but the case of a collisionless stellar system can be further analyzed and the theoretical predictions about the formation of stellar clusters can be made. Then, a generalization of the Boltzmann equation and the Jeans instability can be formulated.

Extended theories of gravity under the assumption of hydrostatic equilibrium

The formation and especially the evolution of stars are other tests of the validity of ETGS and of their compatibility with current knowledge. Observed stellar structures are incompatible with the standard models of stellar structure. In particular, we refer to the neutron stars (magnetars) with mass larger than the Volkoff mass. It therefore seems that on particular length scales the gravitational force is larger or smaller than the corresponding value in GR. For example, a modification of the Hilbert-Einstein Lagrangian consisting of an R^2 term enables a major attraction while a R_{\alpha\beta} R^{\alpha\beta} term gives a repulsive contribution. Understanding on which scales the modifications to GR are activated or what is the weight of corrections to gravitational potential is a crucial point.

Massive states of gravitational waves in extended theories of gravity

A relevant aspect of higher order gravity theories is that, in the post-Minkowskian limit (i.e., with weak fields and arbitrary velocities), the propagation of the gravitational fields turns out to be characterized by waves with both tensorial and scalar modes. This feature represents a striking difference between GR and extended gravity since, in the standard Einstein scheme, only tensorial degrees of freedom are allowed. Gravitational waves represent a fundamental tool to discriminate between GR and alternative gravities. A graviton with non-zero mass produces several effects, such as extra degrees of polarization of gravitational wave modes and a frequency-dependent speed of propagation resulting in a non-trivial dispersion relation. Then, the graviton mass could be constrained using future observations of gravitational waves with the LIGO, Virgo, and the space-based LISA experiments. For example, dynamical binary systems emitting gravitational waves could constitute a good testbed to probe massive gravitons. During their dynamical evolution, the frequency of the binary orbit increases, ramping up rapidly in the late stages of the evolution just before coalescence. Then one proposes a systematic study of gravitational waves for a more general fourth order theory (with the addition of all curvature invariants). Such theories should be analyzed for given sources (for example binary systems) by building models of gravitational emission also for further massive modes of propagation. Today the propagation is studied as a perturbation of Minkowski space (weak gravitational waves), while we can investigate also what the propagation on a curved background (“hard” gravitational waves).

Energy and conservation laws in non-linear (extended) theories of gravity

As seen above, specific tools have been developed to investigate and solve related problems in different areas: weak-field and Newtonian limit, symmetries, Noether’s method, etc. usually based on higher order (non-linear) mathematical models. This proposal intends to open up a new pathway aimed to studying fourth or higher order gravity in the general context of gravitational theories (higher-order in a purely metric and/or non-linear framework, or first-order, with both metric and connection variables - `a la Palatini) in order to allow a more general and unifying treatment of many of the aforementioned topics. Specific attention will be devoted to investigations of the Hamiltonian structure of non-linear (higher order) theories of gravity, with the goal of discussing a more general theory and providing specific examples of conservation laws for exact solutions of these theories. Solutions with singularities (with or without horizons, both standard or generalized) and especially black hole solutions will be approached in order to better understand their entropic properties in presence of standard and dark mass-energy. Another main topic, in the framework of exact approach, is Noether Symmetries and Janis Newman method. By composing two methods it is possible to find axially symmetric solutions starting spherically symmetric ones. Since a real tests of ETGs are also the binary systems (to test the theories at higher energy) and when we consider the physics of rotating systems we must consider a specetime with symmetric properties like Kerr metric, the study of axially symmetric solution is a crucial point.



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