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The principal subject of this thesis is the gravitational two-body problem in the extreme-mass-ratio regime---that is, where one mass is significantly smaller than the other---in the full context of our contemporary theory of gravity,…
We analyze the gravitational waves propagation in metric $f(R)$ theories of gravity. In particular, adopting a gauge invariant formalism we clearly determine the exact propagating degrees of freedom. Then, investigating their effects on…
We study hyperbolic systems of one-dimensional partial differential equations under general, possibly non-local boundary conditions. A large class of evolution equations, either on individual 1-dimensional intervals or on general networks,…
Modifications of Einstein's theory of gravitation have been extensively considered in the past years, in connection to both cosmology and quantum gravity. Higher-curvature and higher-derivative gravity theories constitute the main examples…
An approximate strategy for studying the evolution of binary systems of extended objects is introduced. The stars are assumed to be polytropic ellipsoids. The surfaces of constant density maintain their ellipsoidal shape during the time…
There is sufficient amount of internal evidence in the nature of gravitational theories to indicate that gravity is an emergent phenomenon like, e.g, elasticity. Such an emergent nature is most apparent in the structure of gravitational…
A unified general approach is presented for construction of solutions of the characteristic initial value problems for various integrable hyperbolic reductions of Einstein's equations for space-times with two commuting isometries in General…
We perform a numerical free evolution of a selfgravitating, spherically symmetric scalar field satisfying the wave equation. The evolution equations can be written in a very simple form and are symmetric hyperbolic in Eddington-Finkelstein…
By using geometric methods and superenergy tensors, we find new simple criteria for the causal propagation of physical fields in spacetimes of any dimension. The method can be applied easily to many different theories and to arbitrary…
We develop a general formalism for treating radiative degrees of freedom near $\mathscr{I}^{+}$ in theories with an arbitrary Ricci-flat internal space. These radiative modes are encoded in a generalized news tensor which decomposes into…
Bonazzola, Gourgoulhon, Grandcl\'ement, and Novak [Phys. Rev. D {\bf 70}, 104007 (2004)] proposed a new formulation for 3+1 numerical relativity. Einstein equations result, according to that formalism, in a coupled elliptic-hyperbolic…
A new numerical framework, based on the use of a simple first order strongly hyperbolic evolution equations, is introduced and tested in case of 4-dimensional spherically symmetric gravitating systems. The analytic setup is chosen such that…
When constructing general relativity (GR), Einstein required 4D general covariance. In contrast, we derive GR (in the compact, without boundary case) as a theory of evolving 3-dimensional conformal Riemannian geometries obtained by imposing…
It is argued in the literature that gravity is an emergent phenomenon and is a statistical tendency for a gravitational system to attain the maximal entropy state which maintains the holographic principle. In this paper, we show that…
Solving the 4-d Einstein equations as evolution in time requires solving equations of two types: the four elliptic initial data (constraint) equations, followed by the six second order evolution equations. Analytically the constraint…
The spectral fluctuations of a quantum Hamiltonian system with time-reversal symmetry are studied in the semiclassical limit by using periodic-orbit theory. It is found that, if long periodic orbits are hyperbolic and uniformly distributed…
The Generalized Riemann Problems (GRP) for nonlinear hyperbolic systems of balance laws in one space dimension are now well-known and can be formulated as follows: Given initial-data which are smooth on two sides of a discontinuity,…
The constraint equations for smooth $[n+1]$-dimensional (with $n\geq 3$) Riemannian or Lorentzian spaces satisfying the Einstein field equations are considered. It is shown, regardless of the signature of the primary space, that the…
A new approach to gravitational gauge-invariant perturbation theory begins from the fourth-order Einstein-Ricci system, a hyperbolic formulation of gravity for arbitrary lapse and shift whose centerpiece is a wave equation for curvature. In…
We give a rigorous and mathematically clear presentation of the Covariant and Gauge Invariant theory of gravitational waves in a perturbed Friedmann-Lemaitre-Robertson-Walker universe for Fourth Order Gravity, where the matter is described…