We review the experimental evidence for Einstein's special and general relativity. A variety of high precision null experimentsverify the weak equivalence principle and local Lorentz invariance,while gravitational redshift and other clock experiments support localposition invariance. Together these results confirm theEinstein Equivalence Principle which underlies the concept thatgravitation must bedescribed by a metric theory.Solar system experiments that test the weak-field, post-Newtonianlimit of metric theories strongly favorgeneral relativity. The BinaryPulsar provides tests of gravitational-wave dampingand of strong-field general relativity. Recently discovered binarypulsar systems may provide additional tests. Ongoing and futureexperiments, such as the Gravity Probe B GyroscopeExperiment, satellite tests of the Equivalence principle, and tests ofgravity at short distance to look for extra spatial dimensions could constrain extensions of general relativity.Laser interferometricgravitational-wave observatories on Earth and in space may provide new testsof gravitational theory via detailed measurements of the properties of gravitational waves.

The search for gravitational waves has reached a new era as currently a network of interferometric gravitational wave detectors is in operation around the globe at its initial target sensitivity, in parallel with existing acoustic bar-type detectors, and searches are underway aimed at the first direct detection of gravitational radiation from astrophysical sources. This paper briefly summarizes the status of operating gravitational wave facilities, plans for future detector upgrades and the status of the space-based gravitational wave detector LISA. It also describes some of the most promising sources of gravitational waves for ground-based detectors like LIGO and VIRGO and LISA.

Gamma-Ray Bursts (GRBS) are the brightest explosions observed. Aftera serendipitous discovery and forty years of search we begin tounderstand their origin (type Ic supernovae for long bursts andneutron star mergers for short ones) and how they operate. While theresearch on the nature of GRBs still goes on we turn to applicationof GRBs to study other phenomena. GRBs have an enormous potential toexplore new regimes of ultra-relativistic flows and of very highmagnetic fields. A lot have been said about the potential usage ofGRBs to determine the conditions at the early universe. Here wediscuss an application of GRBs that involve new physics – namely,using the arrival times of photons and neutrinos at differentenergies to set limits on Lorentz invariance violation that couldarise from quantum gravity effects.

I summarize some of the observational phenomenology in compact X-raysources (neutron stars, stellar-mass black holes and super-massiveblack holes) where X-ray observations reveal strong gravity inaction. Although there is no attempt to claim that theseobservations constitute or might lead in the future to real tests ofGeneral Relativity, they allow us to witness a variety of itseffects in the strong field limit.

Gravitational lenses yield a very high rate of return onobservational investment. Given their scarcity, their impact on ourknowledge of the universe is very significant. In the weak-fieldlimit, lensing studies are based on well-established physics and thusoffer a straightforward approach to pursue many currently pressingproblems of astrophysics and cosmology. Examples of these are thesignificance of dark matter and the density, age and size of theuniverse. I present recent developments in cosmological applicationsof gravitational lenses, focusing on estimates of the Hubble constantusing strong lensing of quasars. I describe our recent measurements oftime delays for the images of SDSS J1004+4112, and discuss prospects forthe future utilizing synoptic telescopes, planned and underconstruction.

The harmonic formulation of Einstein's field equations isconsidered, where the gauge conditions are introduced as dynamicalconstraints. The difference between the fully constrained approach(used in analytical approximations) and the free evolution one(used in most numerical approximations) is pointed out. As ageneralization, quasi-stationary gauge conditions are alsodiscussed, including numerical experiments with the gauge-wavestestbed. The complementary 3+1 approach is also considered, whereconstraints are related instead with energy and momentum firstintegrals and the gauge must be provided separately. Therelationship between the two formalisms is discussed in a moregeneral framework (Z4 formalism). Different strategies in blackhole simulations follow when introducing singularity avoidance asa requirement. More flexible quasi-stationary gauge conditions areproposed in this context, which can be seen as generalizations ofthe current “freezing shift” prescriptions.

Interacting quintessence models have been proposed to explain or,at least alleviate, the coincidence problem of late cosmicacceleration. We show that it is possible to construct models thatgive rise to three different epochs: radiation, matter andaccelerated, lasting long enough to allow the emergence of largescale structure. We confront our model with data on CosmicMicrowave Background temperature anisotropies, matter powerspectrum and Supernovae type Ia luminosity distances and find thatthe data are only capable of setting upper limits on theinteraction. We show how the integrated Sachs-Wolfe effect couldprovide indirect evidence of the existence of the interaction but,at present, the prospects that the coincidence problem can beexplained by the coupling of dark matter to dark energy are notstatistically favored by the data.

Core-collapse supernovae have been supposed tobe one of the most plausible sources of gravitational waves. Based on a series of our magnetohydrodynamic core-collapsesimulations, we find that the gravitational amplitudes at core bounce canbe within the detection limits for the currently runninglaser-interferometers for a galactic supernova if the central corerotates sufficiently rapidly. This is regardless of the difference of the realisticequations of state and the possible occurrence of the QCD phasetransition near core bounce. Even if the core rotates slowly, we pointout that thegravitational waves generated from anisotropic neutrino radiationin the postbounce phase due to the standing accretion shockinstability (SASI) could be within the detection limits of the detectors in the next generation such as LCGT and the advanced LIGO for the galactic source.Since the waveforms significantly depend on the exploding scenarios,our results suggest that we can obtain the informationabout the long-veiled explosion mechanism from the gravitational wave signals.Furthermore we discuss the gravitational wave background (GWB) from theexplosions of Pop III stars and show that the GWB from Pop III,depending on their formation rates, can be large enough to be within the detection limitsof future planned interferometers such asDECIGO and BBO in the frequency interval of ~0.1-1 Hz.This means that the detections of GW background from Pop III stars canbe an important tool to supplythe information about the star formation history in the early universe.

A new type of singularity theorem, based on spatial averages of physical quantities, is presented and discussed. Alternatively, the results inform us of when a spacetime can be singularity-free. This theorem provides a decisive observational difference between singular and non-singular, globally hyperbolic, open cosmological models.

Following an earlier work (Tod 2007a), we consider static and stationary solutions ofthe Einstein-Maxwell equations in whichthe Maxwell field is not constant along the Killing vector definingtime-independence, so that the symmetry of the space-time is notinherited by the electromagnetic field. In particular, we look fornoninheriting black hole solutions and find severe constraintson the possibilities.

We study the behaviour of an initially spherical bunch of particlesemitted along trajectories parallel to the symmetry axis of a Kerr black hole. We show that, under suitable conditions, curvature and inertial strains compete to generate jet-like structures.

The theory of location systems involves the geometric and physicaldescription of the protocols allowing the realization of coordinatesystems. In this communication, the incidence of the space-timecausal structure (Newtonian or relativistic) on the construction oflocation systems is remarked. Specifically, we focus our attention:(i) on the construction of Newtonian emission coordinates that arecontrasted with those associated with relativistic positioningsystems, and (ii) on the role played by non-absolutesynchronizations (like the one provided by the local Solar time) inthe comprehension of Newtonian and relativistic location systems.

The trajectory of a charged-massive point-particle in presence of an external magnetic field, at the classical level, is here obtained by using spinorial equations of motion. Within this representation the solution is given by means of two 2-component Weyl-spinors which contain global phase-factors coinciding with the quantum representation of a 1/2 spin particle in its rest-frame. Each individual spinor may be then associated to the up- and -down state respectively. The formalism then appears to reveal classical properties of the spin as well as quantum ones.

FLRW models filled with just dark energy are shown to have a finitepast, since causal geodesics cannot be extended beyond a certainproper time. It is shown that curvature measured along causalgeodesics becomes infinity on travelling to the past, though curvature scalars tend to zero. Furthermore the time measured by free-falling observers from coincidence time to Big Rip is shown to be as short as wished by increasing their linear momentum.

We summarize our studies on properties of double neutron starsobservable in gravitational waves and in the radio. Binary populationsynthesis is used to calculate expected masses of neutron stars inbinary systems taking into account the selection efects specific to each method of detection.We discuss recent relativistic results on final phaseof inspiral of neutron star binaries showing how we can imposeconstraints on the equation of state of dense nuclear matter withgravitational wave observations.

We present the Covariant and Gauge Invariant theory of scalar perturbations of a Friedmann-Lemaître-Robertson-Walker universe for Fourth Order Gravity, where the matter is described by a perfect fluid with a barotropic equation of state. The general perturbation equations are applied to a simple background solution of Rn gravity and we obtain exact solutions of the perturbation equations for scales much bigger than the Hubble radius. These solutions have a number of interesting features. In particular we find that for all values of n there is always a growing mode for the density contrast, even if the universe is in a state of accelerated expansion. Such a behavior does not occur in standard General Relativity, where as soon as Dark Energy dominates, the density contrast experiences an unrelenting decay. Consequently, this peculiarity in the framework of higher order gravity model is sufficently novel to warrant further investigation.

It was Zel'dovich who first clearly realized the importance ofquantum vacuum fluctuations in connection with cosmology. Sincethen, this issue has been more a problem than a solution, even ifthe acceleration of the universe expansion could, in principle, beeasily explained by resorting to it. We have calculated thosecontributions in a number of different models. Under some specificconditions, the results look promising, although a generalacceptable solution looks still far from reach.

We constrain a set of general relativistic dark energy models, togetherwith some other models motivated by extra dimensions, using a combination of three background geometry tests: H(z) data, CMB shift and SNIa data. Resting mostly on Bayesianstatistics, we conclude that LCDM is at least substantially favored, and that models with extra dimensional motivation are less favored than general relativistic ones.

We find that deviation from the virial equilibrium of the Abell Cluster A586 yields evidence of the interaction between dark matter and dark energy. We argue that this interaction might imply a violation of the Equivalence Principle. These evidences are found in the context of two different models of dark energy-dark matter interaction.