Showing papers on "Geodesic deviation published in 2004"

Cosmology in Scalar-Tensor Gravity

Abstract: 1. Scalar-Tensor Gravity.- 1 Introduction.- 2 Brans-Dicke theory.- 3 Brans-Dicke cosmology in the Jordan frame.- 4 The limit to general relativity.- 5 Relation to Kaluza-Klein theory.- 6 Brans-Dicke theory from Lyra's geometry.- 7 Scalar-tensor theories.- 7.1 Effective Lagrangians and Hamiltonians.- 8 Motivations for scalar-tensor theories.- 9 Induced gravity.- 10 Generalized scalar-tensor theories.- 11 Conformal transformation techniques.- 11.1 Conformal transformations.- 11.2 Brans-Dicke theory.- 11.3 Kaluza-Klein cosmology.- 11.4 Scalar-tensor theories.- 11.5 Generalized scalar-tensor theories.- 12 Singularities of the gravitational coupling.- 2. Effective Energy-Momentum Tensors and Conformal Frames.- 1 The issue of the conformal frame.- 1.1 The first viewpoint.- 1.2 The second viewpoint.- 1.3 The third viewpoint.- 1.4 Other viewpoints.- 1.5 Einstein frame or Jordan frame?.- 1.6 Energy conditions in relativistic theories.- 1.7 Singularity theorems and energy conditions.- 2 Effective energy-momentum tensors.- 2.1 Time-dependence of the gravitational coupling.- 2.2 Conservation equations for the various Tab(J) [oo].- 3. Gravitational Waves.- 1 Introduction.- 2 Einstein frame scalar-tensor waves.- 2.1 Gravitational waves in the Einstein frame.- 2.2 Corrections to the geodesic deviation equation.- 3 Gravitational lensing by scalar-tensor gravitational waves.- 3.1 Jordan frame analysis.- 3.2 Einstein frame analysis.- 3.3 Propagation of light through a gravitational wave background.- 4. Exact Solutions of Scalar-Tensor Cosmology.- 1 Introduction.- 2 Exact solutions of Brans-Dicke cosmology.- 2.1 K = 0 FLRW solutions.- 2.1.1 The O'Hanlon and Tupper solution.- 2.1.2 The Brans-Dicke dust solution.- 2.1.3 The Nariai solution.- 2.1.4 Other solutions with cosmological constant.- 2.1.5 Generalizing Nariai's solution.- 2.1.6 Phase space analysis for K = 0 and V(o) = 0.- 2.1.7 Phase plane analysis for K = 0 and V(o) = Ao.- 2.2 K = +-1 solutions and phase space for V = 0.- 2.3 Phase space for any K and V = m2o2/2.- 2.3.1 The Dehnen-Obregon solution.- 2.4 Bianchi models.- 2.4.1 Bianchi V universes.- 3 Exact solutions of scalar-tensor theories.- 5. The Early Universe.- 1 Introduction.- 2 Extended inflation.- 2.1 The original extended inflationary scenario.- 2.2 Alternatives.- 3 Hyperextended inflation.- 4 Real inflation?.- 5 Constraints from primordial nucleosynthesis.- 6. Perturbations.- 1 Introduction.- 2 Scalar perturbations.- 3 Tensor perturbations.- 7. Nonminimal Coupling.- 1 Introduction.- 1.1 Generalized inflation.- 1.2 Motivations for nonminimal coupling.- 1.3 Which value of.- 2 Effective energy-momentum tensors.- 2.1 Approach a la Callan-Coleman-Jackiw.- 2.2 Effective coupling.- 2.3 A mixed approach.- 2.4 Discussion.- 2.5 Energy conditions in FLRW cosmology.- 2.6 Nonminimal coupling and gravitational waves.- 3 Conformal transformations.- 4 Inflation and ? 0: the unperturbed universe.- 4.1 Necessary conditions for generalized inflation.- 4.1.1 Specific potentials.- 4.2 The effective equation of state with nonminimal coupling.- 4.3 Critical values of the scalar field.- 5 The slow-roll regime of generalized inflation.- 5.1 Derivation of the stability conditions.- 5.2 Slow-roll parameters.- 6 Inflation and ? 0: perturbations.- 6.1 Density perturbations.- 6.2 Tensor perturbations.- 7 Conclusion.- 8. The Present Universe.- 1 Present acceleration of the universe and quintessence.- 1.1 Coupled quintessence.- 1.2 Multiple field quintessence.- 1.3 Falsifying quintessence models.- 2 Quintessence with nonminimal coupling.- 2.1 Models using the Ratra-Peebles potential.- 2.2 Necessary conditions for accelerated expansion.- 2.3 Doppler peaks with nonminimal coupling.- 3 Superquintessence.- 3.1 An exact superaccelerating solution.- 3.2 Big Smash singularities.- 4 Quintessence in scalar-tensor gravity.- 5 Conclusion.- References.

Geodesic behavior of sudden future singularities

Abstract: In this paper we analyze the effect of recently proposed classes of sudden future singularities on causal geodesics of FLRW spacetimes. Geodesics are shown to be extendible and just the equations for geodesic deviation are singular, although tidal forces are not strong enough to produce a Big Rip. For the sake of completeness, we compare with the typical sudden future singularities of phantom cosmologies.

Weak lensing in generalized gravity theories

Abstract: We extend the theory of weak gravitational lensing to cosmologies with generalized gravity, described in the Lagrangian by a generic function depending on the Ricci scalar and a nonminimal coupled scalar field. We work out the generalized Poisson equations relating the dynamics of the fluctuating components to the two gauge-invariant scalar gravitational potentials, fixing the contributions from the modified background expansion and fluctuations. We show how the lensing equation gets modified by the cosmic expansion as well as by the presence of anisotropic stress, which is non-null at the linear level both in scalar-tensor gravity and in theories where the gravitational Lagrangian term features a nonminimal dependence on the Ricci scalar. Starting from the geodesic deviation, we derive the generalized expressions for the shear tensor and projected lensing potential, encoding the spacetime variation of the effective gravitational constant and isolating the contribution of the anisotropic stress, which introduces a correction due to the spatial correlation between the gravitational potentials. Finally, we work out the expressions of the lensing convergence power spectrum as well as the correlation between the lensing potential and the integrated Sachs-Wolfe effect affecting cosmic microwave background total intensity and polarization anisotropies. To illustrate phenomenologically the effects, we work out approximate expressions for the quantities above in extended quintessence scenarios where the scalar field coupled to gravity plays the role of the dark energy.

Review: Hamiltonian Linearization of the Rest-Frame Instant Form of Tetrad Gravity in a Completely Fixed 3-Orthogonal Gauge: A Radiation Gauge for Background-Independent Gravitational Waves in a Post-Minkowskian Einstein Spacetime

Abstract: In the framework of the rest-frame instant form of tetrad gravity, where the Hamiltonian is the weak ADM energy \(\hat E_{ADM}\), we define a special completely fixed 3-orthogonal Hamiltonian gauge, corresponding to a choice of non-harmonic 4-coordinates, in which the independent degrees of freedom of the gravitational field are described by two pairs of canonically conjugate Dirac observables (DO) \(r_{\overline a } (\tau ,\overrightarrow \sigma ),\pi \overline a (\tau ,\overrightarrow \sigma ),\overline a = 1,2\). We define a Hamiltonian linearization of the theory, i.e. gravitational waves, withoutintroducing any background 4-metric, by retaining only the linear terms in the DO's in the super-hamiltonian constraint (the Lichnerowicz equation for the conformal factor of the 3-metric) and the quadratic terms in the DO's in \(\hat E_{ADM}\). We solve all the constraints of the linearized theory: this amounts to work in a well defined post-Minkowskian Christodoulou-Klainermann space-time. The Hamilton equations imply the wave equation for the DO's \(r_{\overline a } (\tau ,\overrightarrow \sigma ),\), which replace the two polarizations of the TT harmonic gauge, and that linearized Einstein's equations are satisfied. Finally we study the geodesic equation, both for time-like and null geodesics, and the geodesic deviation equation.

Coupling of Linearized Gravity to Nonrelativistic Test Particles: Dynamics in the General Laboratory Frame

Abstract: The coupling of gravity to matter is explored in the linearized gravity limit. The usual derivation of gravity-matter couplings within the quantum-field-theoretic framework is reviewed. A number of inconsistencies between this derivation of the couplings, and the known results of tidal effects on test particles according to classical general relativity are pointed out. As a step towards resolving these inconsistencies, a General Laboratory Frame fixed on the worldline of an observer is constructed. In this frame, the dynamics of nonrelativistic test particles in the linearized gravity limit is studied, and their Hamiltonian dynamics is derived. It is shown that for stationary metrics this Hamiltonian reduces to the usual Hamiltonian for nonrelativistic particles undergoing geodesic motion. For nonstationary metrics with long-wavelength gravitational waves (GWs) present, it reduces to the Hamiltonian for a nonrelativistic particle undergoing geodesic deviation motion. Arbitrary-wavelength GWs couple to the test particle through a vector-potential-like field Na, the net result of the tidal forces that the GW induces in the system, namely, a local velocity field on the system induced by tidal effects as seen by an observer in the general laboratory frame. Effective electric and magnetic fields, which are related to the electric and magnetic parts of the Weyl tensor, are constructed from Na that obey equations of the same form as Maxwell’s equations . A gedankin gravitational Aharonov-Bohm-type experiment using Na to measure the interference of quantum test particles is presented.

On the viability of local criteria for chaos

Abstract: We consider here a recently proposed geometrical criterion for local instability based on the geodesic deviation equation. Although such a criterion can be useful in some cases, we show here that, in general, it is neither necessary nor sufficient for the occurrence of chaos. To this purpose, we introduce a class of chaotic two-dimensional systems with Gaussian curvature everywhere positive and, hence, locally stable. We show explicitly that chaotic behavior arises from some trajectories that reach certain non convex parts of the boundary of the effective Riemannian manifold. Our result questions, once more, the viability of local, curvature-based criteria to predict chaotic behavior.

Differing Calculations of the Response of Matter-wave Interferometers to Gravitational Waves

Abstract: There now exists in the literature two different expressions for the phase shift of a matterwave interferometer caused by the passage of a gravitation wave. The first, a commonly accepted expression that was first derived in the 1970s, is based on the traditional geodesic equation of motion (EOM) for a test particle. The second, a more recently derived expression, is based on the geodesic deviation EOM. The power-law dependence on the frequency of the gravitational wave for both expressions for the phase shift is different, which indicates fundamental differences in the physics on which these calculations are based. Here we compare the two approaches by presenting a series of side-by-side calculations of the phase shift for one specific matter-wave-interferometer configuration that uses atoms as the interfering particle. By looking at the low-frequency limit of the different expressions for the phase shift obtained, we find that the phase shift calculated via the geodesic deviation EOM is correct, and the ones calculated via the geodesic EOM are not.

Generalized Paraxial Ray Trace Procedure Derived from Geodesic Deviation

Abstract: Paraxial ray tracing procedures have become widely accepted techniques for acoustic models in seismology and underwater acoustics. To date a generic form of these procedures including fluid motion and time dependence has not appeared in the literature. A detailed investigation of the characteristic curves of the equations of hydrodynamics allows for an immediate generalization of the procedure to be extracted from the equation form geodesic deviation. The general paraxial ray trace equations serve as an ideal supplement to ordinary ray tracing in predicting the deformation of acoustic beams in random environments. The general procedure is derived in terms of affine parameterization and in a coordinate time parameterization ideal for application to physical acoustic ray propagation. The formalism is applied to layered media, where the deviation equation reduces to a second order differential equation for a single field with a general solution in terms of a depth integral along the ray path. Some features are illustrated through special cases which lead to exact solutions in terms of either ordinary or special functions.

The Relative Motion of Membranes

Abstract: The relative classical motion of membranes is governed by an equation of the form D(hessian D separation)=riemann times separation times momentum. This is a generalization of the geodesic deviation equation and can be derived from a simple lagrangian. Quantum mechanically the picture is less clear. Some quantizations of the classical equations are attempted so that the question as to whether the Universe started with a quantum fluctuation can be addressed.

Test particles behavior in the framework of a lagrangian geometric theory with propagating torsion

Abstract: Working in the lagrangian framework, we develop a geometric theory in vacuum with propagating torsion; the antisymmetric and trace parts of the torsion tensor, considered as derived from local potential fields, are taken and, using the minimal action principle, their field equations are calculated. Actually these will show themselves to be just equations for propagating waves giving torsion a behavior similar to that of metric which, as known, propagates through gravitational waves. Then we establish a principle of minimal substitution to derive test particles equation of motion, obtaining, as result, that they move along autoparallels. We then calculate the analogous of the geodesic deviation for these trajectories and analyze their behavior in the nonrelativistic limit, showing that the torsion trace potential $\phi$ has a phenomenology which is indistinguishable from that of the gravitational newtonian field; in this way we also give a reason for why there have never been evidence for it.

Piecewise-maximum closed Geodesic trajectories on bounded Toroidal manifolds

Abstract: We use toroidal coordinates for the investigation of maximum geodesic trajectories for arbitrary parameters of a torus. Conditions under which trajectories are located in a bounded part of the toroidal manifold are considered. Using global invariants, we construct closed piecewise-maximum geodesic trajectories.

Riemannian geometry of gravity waves turbulent black hole analogs

Abstract: The gravity water wave black (GWBH) hole analog discovered by Schutzhold and Unruh (SU) is extended to allow for the presence of turbulent shear flow. The Riemannian geometry of turbulent black holes (BH) analogs in water waves is computed in the case of laminar tirbulent shear flow. The Riemann curvature is constant and the geodesic deviation equation shows that the curvature acts locally as a diverging lens and the stream lines on opposite sides of the analog black hole flow apart from each other. In this case it is shown that the curvature quantities can be expressed in terms of the Newtonian gravitational constant in the ergoregion. The dispersion relation is obtained for the case of constant flow injection.